Theorem wemaplem3 9571

Description: Lemma for wemapso 9574. 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 9569	. . . 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 582	. . 3 ⊢ (𝜑 → (𝑃𝑇𝑋 ↔ ∃𝑎 ∈ 𝐴 ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)))))
7	1, 6	mpbid 231	. 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 9569	. . . 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 582	. . 3 ⊢ (𝜑 → (𝑋𝑇𝑄 ↔ ∃𝑏 ∈ 𝐴 ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐)))))
12	8, 11	mpbid 231	. 2 ⊢ (𝜑 → ∃𝑏 ∈ 𝐴 ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))
13	2	ad2antrr 724	. . . . 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 724	. . . . 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 724	. . . . 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 724	. . . . 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 724	. . . . 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 775	. . . . 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 1239	. . . . . 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 1115	. . . . 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 771	. . . . . 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 725	. . . . 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 769	. . . . 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 779	. . . . 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 780	. . . . 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 9570	. . . 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 3146	. . 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 3146	. 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 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3051  ∃wrex 3060   class class class wbr 5143  {copab 5205   Po wpo 5582   Or wor 5583  ‘cfv 6543  (class class class)co 7416   ↑_m cmap 8843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-po 5584  df-so 5585  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7991  df-2nd 7992  df-map 8845

This theorem is referenced by:  wemappo  9572