Theorem isocnv 5814

Description: Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)

Assertion

Ref	Expression
isocnv	⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴))

Proof of Theorem isocnv

Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1ocnv 5476	. . . 4 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → ◡𝐻:𝐵–1-1-onto→𝐴)
2	1	adantr 276	. . 3 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) → ◡𝐻:𝐵–1-1-onto→𝐴)
3		f1ocnvfv2 5781	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑧 ∈ 𝐵) → (𝐻‘(◡𝐻‘𝑧)) = 𝑧)
4	3	adantrr 479	. . . . . . 7 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝐻‘(◡𝐻‘𝑧)) = 𝑧)
5		f1ocnvfv2 5781	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑤 ∈ 𝐵) → (𝐻‘(◡𝐻‘𝑤)) = 𝑤)
6	5	adantrl 478	. . . . . . 7 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝐻‘(◡𝐻‘𝑤)) = 𝑤)
7	4, 6	breq12d 4018	. . . . . 6 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘(◡𝐻‘𝑤)) ↔ 𝑧𝑆𝑤))
8	7	adantlr 477	. . . . 5 ⊢ (((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘(◡𝐻‘𝑤)) ↔ 𝑧𝑆𝑤))
9		f1of 5463	. . . . . . 7 ⊢ (◡𝐻:𝐵–1-1-onto→𝐴 → ◡𝐻:𝐵⟶𝐴)
10	1, 9	syl 14	. . . . . 6 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → ◡𝐻:𝐵⟶𝐴)
11		ffvelcdm 5651	. . . . . . . . 9 ⊢ ((◡𝐻:𝐵⟶𝐴 ∧ 𝑧 ∈ 𝐵) → (◡𝐻‘𝑧) ∈ 𝐴)
12		ffvelcdm 5651	. . . . . . . . 9 ⊢ ((◡𝐻:𝐵⟶𝐴 ∧ 𝑤 ∈ 𝐵) → (◡𝐻‘𝑤) ∈ 𝐴)
13	11, 12	anim12dan 600	. . . . . . . 8 ⊢ ((◡𝐻:𝐵⟶𝐴 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((◡𝐻‘𝑧) ∈ 𝐴 ∧ (◡𝐻‘𝑤) ∈ 𝐴))
14		breq1 4008	. . . . . . . . . . 11 ⊢ (𝑥 = (◡𝐻‘𝑧) → (𝑥𝑅𝑦 ↔ (◡𝐻‘𝑧)𝑅𝑦))
15		fveq2 5517	. . . . . . . . . . . 12 ⊢ (𝑥 = (◡𝐻‘𝑧) → (𝐻‘𝑥) = (𝐻‘(◡𝐻‘𝑧)))
16	15	breq1d 4015	. . . . . . . . . . 11 ⊢ (𝑥 = (◡𝐻‘𝑧) → ((𝐻‘𝑥)𝑆(𝐻‘𝑦) ↔ (𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦)))
17	14, 16	bibi12d 235	. . . . . . . . . 10 ⊢ (𝑥 = (◡𝐻‘𝑧) → ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ ((◡𝐻‘𝑧)𝑅𝑦 ↔ (𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦))))
18		bicom 140	. . . . . . . . . 10 ⊢ (((◡𝐻‘𝑧)𝑅𝑦 ↔ (𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦)) ↔ ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦) ↔ (◡𝐻‘𝑧)𝑅𝑦))
19	17, 18	bitrdi 196	. . . . . . . . 9 ⊢ (𝑥 = (◡𝐻‘𝑧) → ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦) ↔ (◡𝐻‘𝑧)𝑅𝑦)))
20		fveq2 5517	. . . . . . . . . . 11 ⊢ (𝑦 = (◡𝐻‘𝑤) → (𝐻‘𝑦) = (𝐻‘(◡𝐻‘𝑤)))
21	20	breq2d 4017	. . . . . . . . . 10 ⊢ (𝑦 = (◡𝐻‘𝑤) → ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦) ↔ (𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘(◡𝐻‘𝑤))))
22		breq2 4009	. . . . . . . . . 10 ⊢ (𝑦 = (◡𝐻‘𝑤) → ((◡𝐻‘𝑧)𝑅𝑦 ↔ (◡𝐻‘𝑧)𝑅(◡𝐻‘𝑤)))
23	21, 22	bibi12d 235	. . . . . . . . 9 ⊢ (𝑦 = (◡𝐻‘𝑤) → (((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦) ↔ (◡𝐻‘𝑧)𝑅𝑦) ↔ ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘(◡𝐻‘𝑤)) ↔ (◡𝐻‘𝑧)𝑅(◡𝐻‘𝑤))))
24	19, 23	rspc2va 2857	. . . . . . . 8 ⊢ ((((◡𝐻‘𝑧) ∈ 𝐴 ∧ (◡𝐻‘𝑤) ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) → ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘(◡𝐻‘𝑤)) ↔ (◡𝐻‘𝑧)𝑅(◡𝐻‘𝑤)))
25	13, 24	sylan 283	. . . . . . 7 ⊢ (((◡𝐻:𝐵⟶𝐴 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) → ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘(◡𝐻‘𝑤)) ↔ (◡𝐻‘𝑧)𝑅(◡𝐻‘𝑤)))
26	25	an32s 568	. . . . . 6 ⊢ (((◡𝐻:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘(◡𝐻‘𝑤)) ↔ (◡𝐻‘𝑧)𝑅(◡𝐻‘𝑤)))
27	10, 26	sylanl1 402	. . . . 5 ⊢ (((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘(◡𝐻‘𝑤)) ↔ (◡𝐻‘𝑧)𝑅(◡𝐻‘𝑤)))
28	8, 27	bitr3d 190	. . . 4 ⊢ (((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧𝑆𝑤 ↔ (◡𝐻‘𝑧)𝑅(◡𝐻‘𝑤)))
29	28	ralrimivva 2559	. . 3 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (◡𝐻‘𝑧)𝑅(◡𝐻‘𝑤)))
30	2, 29	jca 306	. 2 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) → (◡𝐻:𝐵–1-1-onto→𝐴 ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (◡𝐻‘𝑧)𝑅(◡𝐻‘𝑤))))
31		df-isom 5227	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
32		df-isom 5227	. 2 ⊢ (◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) ↔ (◡𝐻:𝐵–1-1-onto→𝐴 ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (◡𝐻‘𝑧)𝑅(◡𝐻‘𝑤))))
33	30, 31, 32	3imtr4i 201	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  ∀wral 2455   class class class wbr 4005  ^◡ccnv 4627  ⟶wf 5214  –1-1-onto→wf1o 5217  ‘cfv 5218   Isom wiso 5219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227

This theorem is referenced by:  isores1  5817  isose  5824  isopo  5826  isoso  5828  isoti  7008  infrenegsupex  9596  infxrnegsupex  11273  relogiso  14379