Theorem wemaplem3 9465

Description: Lemma for wemapso 9468. Transitivity. (Contributed by Stefan O'Rear, 17-Jan-2015.) (Revised by AV, 21-Jul-2024.)

Hypotheses

Ref	Expression
wemapso.t	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
wemaplem2.p	⊢ (𝜑 → 𝑃 ∈ (𝐵 ↑m 𝐴))
wemaplem2.x	⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m 𝐴))
wemaplem2.q	⊢ (𝜑 → 𝑄 ∈ (𝐵 ↑m 𝐴))
wemaplem2.r	⊢ (𝜑 → 𝑅 Or 𝐴)
wemaplem2.s	⊢ (𝜑 → 𝑆 Po 𝐵)
wemaplem3.px	⊢ (𝜑 → 𝑃𝑇𝑋)
wemaplem3.xq	⊢ (𝜑 → 𝑋𝑇𝑄)

Assertion

Ref	Expression
wemaplem3	⊢ (𝜑 → 𝑃𝑇𝑄)

Distinct variable groups:   𝑥,𝐵   𝑥,𝑤,𝑦,𝑧,𝑋   𝑤,𝐴,𝑥,𝑦,𝑧   𝑤,𝑃,𝑥,𝑦,𝑧   𝑤,𝑄,𝑥,𝑦,𝑧   𝑤,𝑅,𝑥,𝑦,𝑧   𝑤,𝑆,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝐵(𝑦,𝑧,𝑤)   𝑇(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem wemaplem3

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wemaplem3.px	. . 3 ⊢ (𝜑 → 𝑃𝑇𝑋)
2		wemaplem2.p	. . . 4 ⊢ (𝜑 → 𝑃 ∈ (𝐵 ↑m 𝐴))
3		wemaplem2.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m 𝐴))
4		wemapso.t	. . . . 5 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
5	4	wemaplem1 9463	. . . 4 ⊢ ((𝑃 ∈ (𝐵 ↑m 𝐴) ∧ 𝑋 ∈ (𝐵 ↑m 𝐴)) → (𝑃𝑇𝑋 ↔ ∃𝑎 ∈ 𝐴 ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)))))
6	2, 3, 5	syl2anc 585	. . 3 ⊢ (𝜑 → (𝑃𝑇𝑋 ↔ ∃𝑎 ∈ 𝐴 ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)))))
7	1, 6	mpbid 232	. 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝐴 ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))
8		wemaplem3.xq	. . 3 ⊢ (𝜑 → 𝑋𝑇𝑄)
9		wemaplem2.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝐵 ↑m 𝐴))
10	4	wemaplem1 9463	. . . 4 ⊢ ((𝑋 ∈ (𝐵 ↑m 𝐴) ∧ 𝑄 ∈ (𝐵 ↑m 𝐴)) → (𝑋𝑇𝑄 ↔ ∃𝑏 ∈ 𝐴 ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐)))))
11	3, 9, 10	syl2anc 585	. . 3 ⊢ (𝜑 → (𝑋𝑇𝑄 ↔ ∃𝑏 ∈ 𝐴 ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐)))))
12	8, 11	mpbid 232	. 2 ⊢ (𝜑 → ∃𝑏 ∈ 𝐴 ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))
13	2	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → 𝑃 ∈ (𝐵 ↑m 𝐴))
14	3	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → 𝑋 ∈ (𝐵 ↑m 𝐴))
15	9	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → 𝑄 ∈ (𝐵 ↑m 𝐴))
16		wemaplem2.r	. . . . . 6 ⊢ (𝜑 → 𝑅 Or 𝐴)
17	16	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → 𝑅 Or 𝐴)
18		wemaplem2.s	. . . . . 6 ⊢ (𝜑 → 𝑆 Po 𝐵)
19	18	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → 𝑆 Po 𝐵)
20		simplrl 777	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → 𝑎 ∈ 𝐴)
21		simp2rl 1244	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → (𝑃‘𝑎)𝑆(𝑋‘𝑎))
22	21	3expa 1119	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → (𝑃‘𝑎)𝑆(𝑋‘𝑎))
23		simprr 773	. . . . . 6 ⊢ ((𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)))) → ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)))
24	23	ad2antlr 728	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)))
25		simprl 771	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → 𝑏 ∈ 𝐴)
26		simprrl 781	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → (𝑋‘𝑏)𝑆(𝑄‘𝑏))
27		simprrr 782	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐)))
28	4, 13, 14, 15, 17, 19, 20, 22, 24, 25, 26, 27	wemaplem2 9464	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → 𝑃𝑇𝑄)
29	28	rexlimdvaa 3140	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) → (∃𝑏 ∈ 𝐴 ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))) → 𝑃𝑇𝑄))
30	29	rexlimdvaa 3140	. 2 ⊢ (𝜑 → (∃𝑎 ∈ 𝐴 ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))) → (∃𝑏 ∈ 𝐴 ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))) → 𝑃𝑇𝑄)))
31	7, 12, 30	mp2d 49	1 ⊢ (𝜑 → 𝑃𝑇𝑄)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   class class class wbr 5100  {copab 5162   Po wpo 5538   Or wor 5539  ‘cfv 6500  (class class class)co 7368   ↑_m cmap 8775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-po 5540  df-so 5541  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-map 8777

This theorem is referenced by:  wemappo  9466