Theorem isocnv 7276

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 6786	. . . 4 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → ◡𝐻:𝐵–1-1-onto→𝐴)
2	1	adantr 480	. . 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 7223	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑧 ∈ 𝐵) → (𝐻‘(◡𝐻‘𝑧)) = 𝑧)
4	3	adantrr 717	. . . . . . 7 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝐻‘(◡𝐻‘𝑧)) = 𝑧)
5		f1ocnvfv2 7223	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑤 ∈ 𝐵) → (𝐻‘(◡𝐻‘𝑤)) = 𝑤)
6	5	adantrl 716	. . . . . . 7 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝐻‘(◡𝐻‘𝑤)) = 𝑤)
7	4, 6	breq12d 5111	. . . . . 6 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘(◡𝐻‘𝑤)) ↔ 𝑧𝑆𝑤))
8	7	adantlr 715	. . . . 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 6774	. . . . . . 7 ⊢ (◡𝐻:𝐵–1-1-onto→𝐴 → ◡𝐻:𝐵⟶𝐴)
10	1, 9	syl 17	. . . . . 6 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → ◡𝐻:𝐵⟶𝐴)
11		ffvelcdm 7026	. . . . . . . . 9 ⊢ ((◡𝐻:𝐵⟶𝐴 ∧ 𝑧 ∈ 𝐵) → (◡𝐻‘𝑧) ∈ 𝐴)
12		ffvelcdm 7026	. . . . . . . . 9 ⊢ ((◡𝐻:𝐵⟶𝐴 ∧ 𝑤 ∈ 𝐵) → (◡𝐻‘𝑤) ∈ 𝐴)
13	11, 12	anim12dan 619	. . . . . . . 8 ⊢ ((◡𝐻:𝐵⟶𝐴 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((◡𝐻‘𝑧) ∈ 𝐴 ∧ (◡𝐻‘𝑤) ∈ 𝐴))
14		breq1 5101	. . . . . . . . . . 11 ⊢ (𝑥 = (◡𝐻‘𝑧) → (𝑥𝑅𝑦 ↔ (◡𝐻‘𝑧)𝑅𝑦))
15		fveq2 6834	. . . . . . . . . . . 12 ⊢ (𝑥 = (◡𝐻‘𝑧) → (𝐻‘𝑥) = (𝐻‘(◡𝐻‘𝑧)))
16	15	breq1d 5108	. . . . . . . . . . 11 ⊢ (𝑥 = (◡𝐻‘𝑧) → ((𝐻‘𝑥)𝑆(𝐻‘𝑦) ↔ (𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦)))
17	14, 16	bibi12d 345	. . . . . . . . . 10 ⊢ (𝑥 = (◡𝐻‘𝑧) → ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ ((◡𝐻‘𝑧)𝑅𝑦 ↔ (𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦))))
18		bicom 222	. . . . . . . . . 10 ⊢ (((◡𝐻‘𝑧)𝑅𝑦 ↔ (𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦)) ↔ ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦) ↔ (◡𝐻‘𝑧)𝑅𝑦))
19	17, 18	bitrdi 287	. . . . . . . . 9 ⊢ (𝑥 = (◡𝐻‘𝑧) → ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦) ↔ (◡𝐻‘𝑧)𝑅𝑦)))
20		fveq2 6834	. . . . . . . . . . 11 ⊢ (𝑦 = (◡𝐻‘𝑤) → (𝐻‘𝑦) = (𝐻‘(◡𝐻‘𝑤)))
21	20	breq2d 5110	. . . . . . . . . 10 ⊢ (𝑦 = (◡𝐻‘𝑤) → ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦) ↔ (𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘(◡𝐻‘𝑤))))
22		breq2 5102	. . . . . . . . . 10 ⊢ (𝑦 = (◡𝐻‘𝑤) → ((◡𝐻‘𝑧)𝑅𝑦 ↔ (◡𝐻‘𝑧)𝑅(◡𝐻‘𝑤)))
23	21, 22	bibi12d 345	. . . . . . . . 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 3588	. . . . . . . 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 580	. . . . . . 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 652	. . . . . 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 680	. . . . 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 281	. . . 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 3179	. . 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 511	. 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 6501	. 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 6501	. 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 292	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051   class class class wbr 5098  ^◡ccnv 5623  ⟶wf 6488  –1-1-onto→wf1o 6491  ‘cfv 6492   Isom wiso 6493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501

This theorem is referenced by:  isores1  7280  isofr  7288  isose  7289  isopo  7292  isoso  7294  weisoeq  7301  weisoeq2  7302  fnwelem  8073  oieu  9444  oemapwe  9603  cantnffval2  9604  wemapwe  9606  infxpenlem  9923  fpwwe2lem6  10547  fpwwe2lem8  10549  infrenegsup  12125  ltweuz  13884  fz1isolem  14384  ordthmeo  23746  relogiso  26563  erdsze2lem2  35398  fzisoeu  45544