Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ctiunctlemfo GIF version

Theorem ctiunctlemfo 12140
 Description: Lemma for ctiunct 12141. (Contributed by Jim Kingdon, 28-Oct-2023.)
Hypotheses
Ref Expression
ctiunct.som (𝜑𝑆 ⊆ ω)
ctiunct.sdc (𝜑 → ∀𝑛 ∈ ω DECID 𝑛𝑆)
ctiunct.f (𝜑𝐹:𝑆onto𝐴)
ctiunct.tom ((𝜑𝑥𝐴) → 𝑇 ⊆ ω)
ctiunct.tdc ((𝜑𝑥𝐴) → ∀𝑛 ∈ ω DECID 𝑛𝑇)
ctiunct.g ((𝜑𝑥𝐴) → 𝐺:𝑇onto𝐵)
ctiunct.j (𝜑𝐽:ω–1-1-onto→(ω × ω))
ctiunct.u 𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑧)) ∈ (𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇)}
ctiunct.h 𝐻 = (𝑛𝑈 ↦ ((𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐺‘(2nd ‘(𝐽𝑛))))
ctiunct.nfh 𝑥𝐻
ctiunct.nfu 𝑥𝑈
Assertion
Ref Expression
ctiunctlemfo (𝜑𝐻:𝑈onto 𝑥𝐴 𝐵)
Distinct variable groups:   𝐴,𝑛,𝑥   𝐵,𝑛   𝑛,𝐹,𝑥,𝑧   𝑛,𝐺   𝑛,𝐽,𝑥,𝑧   𝑧,𝑆   𝑧,𝑇   𝑈,𝑛   𝜑,𝑛,𝑥   𝑧,𝑛
Allowed substitution hints:   𝜑(𝑧)   𝐴(𝑧)   𝐵(𝑥,𝑧)   𝑆(𝑥,𝑛)   𝑇(𝑥,𝑛)   𝑈(𝑥,𝑧)   𝐺(𝑥,𝑧)   𝐻(𝑥,𝑧,𝑛)

Proof of Theorem ctiunctlemfo
Dummy variables 𝑟 𝑤 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ctiunct.som . . 3 (𝜑𝑆 ⊆ ω)
2 ctiunct.sdc . . 3 (𝜑 → ∀𝑛 ∈ ω DECID 𝑛𝑆)
3 ctiunct.f . . 3 (𝜑𝐹:𝑆onto𝐴)
4 ctiunct.tom . . 3 ((𝜑𝑥𝐴) → 𝑇 ⊆ ω)
5 ctiunct.tdc . . 3 ((𝜑𝑥𝐴) → ∀𝑛 ∈ ω DECID 𝑛𝑇)
6 ctiunct.g . . 3 ((𝜑𝑥𝐴) → 𝐺:𝑇onto𝐵)
7 ctiunct.j . . 3 (𝜑𝐽:ω–1-1-onto→(ω × ω))
8 ctiunct.u . . 3 𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑧)) ∈ (𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇)}
9 ctiunct.h . . 3 𝐻 = (𝑛𝑈 ↦ ((𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐺‘(2nd ‘(𝐽𝑛))))
101, 2, 3, 4, 5, 6, 7, 8, 9ctiunctlemf 12139 . 2 (𝜑𝐻:𝑈 𝑥𝐴 𝐵)
11 nfv 1508 . . . . 5 𝑥𝜑
12 nfiu1 3879 . . . . . 6 𝑥 𝑥𝐴 𝐵
1312nfcri 2293 . . . . 5 𝑥 𝑢 𝑥𝐴 𝐵
1411, 13nfan 1545 . . . 4 𝑥(𝜑𝑢 𝑥𝐴 𝐵)
15 ctiunct.nfu . . . . 5 𝑥𝑈
16 ctiunct.nfh . . . . . . 7 𝑥𝐻
17 nfcv 2299 . . . . . . 7 𝑥𝑣
1816, 17nffv 5475 . . . . . 6 𝑥(𝐻𝑣)
1918nfeq2 2311 . . . . 5 𝑥 𝑢 = (𝐻𝑣)
2015, 19nfrexxy 2496 . . . 4 𝑥𝑣𝑈 𝑢 = (𝐻𝑣)
21 simplll 523 . . . . . . 7 ((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) → 𝜑)
22 simplr 520 . . . . . . 7 ((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) → 𝑥𝐴)
2321, 22, 6syl2anc 409 . . . . . 6 ((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) → 𝐺:𝑇onto𝐵)
24 foelrn 5698 . . . . . 6 ((𝐺:𝑇onto𝐵𝑢𝐵) → ∃𝑤𝑇 𝑢 = (𝐺𝑤))
2523, 24sylancom 417 . . . . 5 ((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) → ∃𝑤𝑇 𝑢 = (𝐺𝑤))
263ad4antr 486 . . . . . . 7 (((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) → 𝐹:𝑆onto𝐴)
27 simpllr 524 . . . . . . 7 (((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) → 𝑥𝐴)
28 foelrn 5698 . . . . . . 7 ((𝐹:𝑆onto𝐴𝑥𝐴) → ∃𝑟𝑆 𝑥 = (𝐹𝑟))
2926, 27, 28syl2anc 409 . . . . . 6 (((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) → ∃𝑟𝑆 𝑥 = (𝐹𝑟))
30 f1ocnv 5424 . . . . . . . . . . . 12 (𝐽:ω–1-1-onto→(ω × ω) → 𝐽:(ω × ω)–1-1-onto→ω)
317, 30syl 14 . . . . . . . . . . 11 (𝜑𝐽:(ω × ω)–1-1-onto→ω)
32 f1of 5411 . . . . . . . . . . 11 (𝐽:(ω × ω)–1-1-onto→ω → 𝐽:(ω × ω)⟶ω)
3331, 32syl 14 . . . . . . . . . 10 (𝜑𝐽:(ω × ω)⟶ω)
3433ad5antr 488 . . . . . . . . 9 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → 𝐽:(ω × ω)⟶ω)
351ad5antr 488 . . . . . . . . . . 11 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → 𝑆 ⊆ ω)
36 simprl 521 . . . . . . . . . . 11 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → 𝑟𝑆)
3735, 36sseldd 3129 . . . . . . . . . 10 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → 𝑟 ∈ ω)
38 simp-5l 533 . . . . . . . . . . . 12 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → 𝜑)
3927adantr 274 . . . . . . . . . . . 12 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → 𝑥𝐴)
4038, 39, 4syl2anc 409 . . . . . . . . . . 11 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → 𝑇 ⊆ ω)
41 simplrl 525 . . . . . . . . . . 11 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → 𝑤𝑇)
4240, 41sseldd 3129 . . . . . . . . . 10 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → 𝑤 ∈ ω)
4337, 42opelxpd 4616 . . . . . . . . 9 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → ⟨𝑟, 𝑤⟩ ∈ (ω × ω))
4434, 43ffvelrnd 5600 . . . . . . . 8 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → (𝐽‘⟨𝑟, 𝑤⟩) ∈ ω)
4538, 7syl 14 . . . . . . . . . . . . 13 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → 𝐽:ω–1-1-onto→(ω × ω))
46 f1ocnvfv2 5723 . . . . . . . . . . . . 13 ((𝐽:ω–1-1-onto→(ω × ω) ∧ ⟨𝑟, 𝑤⟩ ∈ (ω × ω)) → (𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)) = ⟨𝑟, 𝑤⟩)
4745, 43, 46syl2anc 409 . . . . . . . . . . . 12 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → (𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)) = ⟨𝑟, 𝑤⟩)
4847fveq2d 5469 . . . . . . . . . . 11 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → (1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩))) = (1st ‘⟨𝑟, 𝑤⟩))
49 vex 2715 . . . . . . . . . . . 12 𝑟 ∈ V
50 vex 2715 . . . . . . . . . . . 12 𝑤 ∈ V
5149, 50op1st 6088 . . . . . . . . . . 11 (1st ‘⟨𝑟, 𝑤⟩) = 𝑟
5248, 51eqtrdi 2206 . . . . . . . . . 10 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → (1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩))) = 𝑟)
5352, 36eqeltrd 2234 . . . . . . . . 9 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → (1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩))) ∈ 𝑆)
5447fveq2d 5469 . . . . . . . . . . 11 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → (2nd ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩))) = (2nd ‘⟨𝑟, 𝑤⟩))
5549, 50op2nd 6089 . . . . . . . . . . 11 (2nd ‘⟨𝑟, 𝑤⟩) = 𝑤
5654, 55eqtrdi 2206 . . . . . . . . . 10 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → (2nd ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩))) = 𝑤)
5752fveq2d 5469 . . . . . . . . . . . . 13 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → (𝐹‘(1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)))) = (𝐹𝑟))
58 simprr 522 . . . . . . . . . . . . 13 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → 𝑥 = (𝐹𝑟))
5957, 58eqtr4d 2193 . . . . . . . . . . . 12 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → (𝐹‘(1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)))) = 𝑥)
6059csbeq1d 3038 . . . . . . . . . . 11 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → (𝐹‘(1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)))) / 𝑥𝑇 = 𝑥 / 𝑥𝑇)
61 csbid 3039 . . . . . . . . . . 11 𝑥 / 𝑥𝑇 = 𝑇
6260, 61eqtrdi 2206 . . . . . . . . . 10 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → (𝐹‘(1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)))) / 𝑥𝑇 = 𝑇)
6341, 56, 623eltr4d 2241 . . . . . . . . 9 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → (2nd ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩))) ∈ (𝐹‘(1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)))) / 𝑥𝑇)
6453, 63jca 304 . . . . . . . 8 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → ((1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩))) ∈ 𝑆 ∧ (2nd ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩))) ∈ (𝐹‘(1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)))) / 𝑥𝑇))
65 2fveq3 5470 . . . . . . . . . . 11 (𝑧 = (𝐽‘⟨𝑟, 𝑤⟩) → (1st ‘(𝐽𝑧)) = (1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩))))
6665eleq1d 2226 . . . . . . . . . 10 (𝑧 = (𝐽‘⟨𝑟, 𝑤⟩) → ((1st ‘(𝐽𝑧)) ∈ 𝑆 ↔ (1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩))) ∈ 𝑆))
67 2fveq3 5470 . . . . . . . . . . 11 (𝑧 = (𝐽‘⟨𝑟, 𝑤⟩) → (2nd ‘(𝐽𝑧)) = (2nd ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩))))
6865fveq2d 5469 . . . . . . . . . . . 12 (𝑧 = (𝐽‘⟨𝑟, 𝑤⟩) → (𝐹‘(1st ‘(𝐽𝑧))) = (𝐹‘(1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)))))
6968csbeq1d 3038 . . . . . . . . . . 11 (𝑧 = (𝐽‘⟨𝑟, 𝑤⟩) → (𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇 = (𝐹‘(1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)))) / 𝑥𝑇)
7067, 69eleq12d 2228 . . . . . . . . . 10 (𝑧 = (𝐽‘⟨𝑟, 𝑤⟩) → ((2nd ‘(𝐽𝑧)) ∈ (𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇 ↔ (2nd ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩))) ∈ (𝐹‘(1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)))) / 𝑥𝑇))
7166, 70anbi12d 465 . . . . . . . . 9 (𝑧 = (𝐽‘⟨𝑟, 𝑤⟩) → (((1st ‘(𝐽𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑧)) ∈ (𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇) ↔ ((1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩))) ∈ 𝑆 ∧ (2nd ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩))) ∈ (𝐹‘(1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)))) / 𝑥𝑇)))
7271, 8elrab2 2871 . . . . . . . 8 ((𝐽‘⟨𝑟, 𝑤⟩) ∈ 𝑈 ↔ ((𝐽‘⟨𝑟, 𝑤⟩) ∈ ω ∧ ((1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩))) ∈ 𝑆 ∧ (2nd ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩))) ∈ (𝐹‘(1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)))) / 𝑥𝑇)))
7344, 64, 72sylanbrc 414 . . . . . . 7 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → (𝐽‘⟨𝑟, 𝑤⟩) ∈ 𝑈)
7459csbeq1d 3038 . . . . . . . . . 10 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → (𝐹‘(1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)))) / 𝑥𝐺 = 𝑥 / 𝑥𝐺)
75 csbid 3039 . . . . . . . . . 10 𝑥 / 𝑥𝐺 = 𝐺
7674, 75eqtrdi 2206 . . . . . . . . 9 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → (𝐹‘(1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)))) / 𝑥𝐺 = 𝐺)
7776, 56fveq12d 5472 . . . . . . . 8 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → ((𝐹‘(1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)))) / 𝑥𝐺‘(2nd ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)))) = (𝐺𝑤))
78 2fveq3 5470 . . . . . . . . . . . 12 (𝑛 = (𝐽‘⟨𝑟, 𝑤⟩) → (1st ‘(𝐽𝑛)) = (1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩))))
7978fveq2d 5469 . . . . . . . . . . 11 (𝑛 = (𝐽‘⟨𝑟, 𝑤⟩) → (𝐹‘(1st ‘(𝐽𝑛))) = (𝐹‘(1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)))))
8079csbeq1d 3038 . . . . . . . . . 10 (𝑛 = (𝐽‘⟨𝑟, 𝑤⟩) → (𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐺 = (𝐹‘(1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)))) / 𝑥𝐺)
81 2fveq3 5470 . . . . . . . . . 10 (𝑛 = (𝐽‘⟨𝑟, 𝑤⟩) → (2nd ‘(𝐽𝑛)) = (2nd ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩))))
8280, 81fveq12d 5472 . . . . . . . . 9 (𝑛 = (𝐽‘⟨𝑟, 𝑤⟩) → ((𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐺‘(2nd ‘(𝐽𝑛))) = ((𝐹‘(1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)))) / 𝑥𝐺‘(2nd ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)))))
83 simplrr 526 . . . . . . . . . . 11 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → 𝑢 = (𝐺𝑤))
8483, 77eqtr4d 2193 . . . . . . . . . 10 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → 𝑢 = ((𝐹‘(1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)))) / 𝑥𝐺‘(2nd ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)))))
85 simpllr 524 . . . . . . . . . 10 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → 𝑢𝐵)
8684, 85eqeltrrd 2235 . . . . . . . . 9 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → ((𝐹‘(1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)))) / 𝑥𝐺‘(2nd ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)))) ∈ 𝐵)
879, 82, 73, 86fvmptd3 5558 . . . . . . . 8 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → (𝐻‘(𝐽‘⟨𝑟, 𝑤⟩)) = ((𝐹‘(1st ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)))) / 𝑥𝐺‘(2nd ‘(𝐽‘(𝐽‘⟨𝑟, 𝑤⟩)))))
8877, 87, 833eqtr4rd 2201 . . . . . . 7 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → 𝑢 = (𝐻‘(𝐽‘⟨𝑟, 𝑤⟩)))
89 fveq2 5465 . . . . . . . 8 (𝑣 = (𝐽‘⟨𝑟, 𝑤⟩) → (𝐻𝑣) = (𝐻‘(𝐽‘⟨𝑟, 𝑤⟩)))
9089rspceeqv 2834 . . . . . . 7 (((𝐽‘⟨𝑟, 𝑤⟩) ∈ 𝑈𝑢 = (𝐻‘(𝐽‘⟨𝑟, 𝑤⟩))) → ∃𝑣𝑈 𝑢 = (𝐻𝑣))
9173, 88, 90syl2anc 409 . . . . . 6 ((((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) ∧ (𝑟𝑆𝑥 = (𝐹𝑟))) → ∃𝑣𝑈 𝑢 = (𝐻𝑣))
9229, 91rexlimddv 2579 . . . . 5 (((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) ∧ (𝑤𝑇𝑢 = (𝐺𝑤))) → ∃𝑣𝑈 𝑢 = (𝐻𝑣))
9325, 92rexlimddv 2579 . . . 4 ((((𝜑𝑢 𝑥𝐴 𝐵) ∧ 𝑥𝐴) ∧ 𝑢𝐵) → ∃𝑣𝑈 𝑢 = (𝐻𝑣))
94 eliun 3853 . . . . . 6 (𝑢 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑢𝐵)
9594biimpi 119 . . . . 5 (𝑢 𝑥𝐴 𝐵 → ∃𝑥𝐴 𝑢𝐵)
9695adantl 275 . . . 4 ((𝜑𝑢 𝑥𝐴 𝐵) → ∃𝑥𝐴 𝑢𝐵)
9714, 20, 93, 96r19.29af2 2597 . . 3 ((𝜑𝑢 𝑥𝐴 𝐵) → ∃𝑣𝑈 𝑢 = (𝐻𝑣))
9897ralrimiva 2530 . 2 (𝜑 → ∀𝑢 𝑥𝐴 𝐵𝑣𝑈 𝑢 = (𝐻𝑣))
99 dffo3 5611 . 2 (𝐻:𝑈onto 𝑥𝐴 𝐵 ↔ (𝐻:𝑈 𝑥𝐴 𝐵 ∧ ∀𝑢 𝑥𝐴 𝐵𝑣𝑈 𝑢 = (𝐻𝑣)))
10010, 98, 99sylanbrc 414 1 (𝜑𝐻:𝑈onto 𝑥𝐴 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  DECID wdc 820   = wceq 1335   ∈ wcel 2128  Ⅎwnfc 2286  ∀wral 2435  ∃wrex 2436  {crab 2439  ⦋csb 3031   ⊆ wss 3102  ⟨cop 3563  ∪ ciun 3849   ↦ cmpt 4025  ωcom 4547   × cxp 4581  ◡ccnv 4582  ⟶wf 5163  –onto→wfo 5165  –1-1-onto→wf1o 5166  ‘cfv 5167  1st c1st 6080  2nd c2nd 6081 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168  ax-un 4392 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4252  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-1st 6082  df-2nd 6083 This theorem is referenced by:  ctiunct  12141
 Copyright terms: Public domain W3C validator