Theorem isocnv 7286

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 6794	. . . 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 7233	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑧 ∈ 𝐵) → (𝐻‘(◡𝐻‘𝑧)) = 𝑧)
4	3	adantrr 718	. . . . . . 7 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝐻‘(◡𝐻‘𝑧)) = 𝑧)
5		f1ocnvfv2 7233	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑤 ∈ 𝐵) → (𝐻‘(◡𝐻‘𝑤)) = 𝑤)
6	5	adantrl 717	. . . . . . 7 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝐻‘(◡𝐻‘𝑤)) = 𝑤)
7	4, 6	breq12d 5113	. . . . . 6 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘(◡𝐻‘𝑤)) ↔ 𝑧𝑆𝑤))
8	7	adantlr 716	. . . . 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 6782	. . . . . . 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 7035	. . . . . . . . 9 ⊢ ((◡𝐻:𝐵⟶𝐴 ∧ 𝑧 ∈ 𝐵) → (◡𝐻‘𝑧) ∈ 𝐴)
12		ffvelcdm 7035	. . . . . . . . 9 ⊢ ((◡𝐻:𝐵⟶𝐴 ∧ 𝑤 ∈ 𝐵) → (◡𝐻‘𝑤) ∈ 𝐴)
13	11, 12	anim12dan 620	. . . . . . . 8 ⊢ ((◡𝐻:𝐵⟶𝐴 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((◡𝐻‘𝑧) ∈ 𝐴 ∧ (◡𝐻‘𝑤) ∈ 𝐴))
14		breq1 5103	. . . . . . . . . . 11 ⊢ (𝑥 = (◡𝐻‘𝑧) → (𝑥𝑅𝑦 ↔ (◡𝐻‘𝑧)𝑅𝑦))
15		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑥 = (◡𝐻‘𝑧) → (𝐻‘𝑥) = (𝐻‘(◡𝐻‘𝑧)))
16	15	breq1d 5110	. . . . . . . . . . 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 6842	. . . . . . . . . . 11 ⊢ (𝑦 = (◡𝐻‘𝑤) → (𝐻‘𝑦) = (𝐻‘(◡𝐻‘𝑤)))
21	20	breq2d 5112	. . . . . . . . . 10 ⊢ (𝑦 = (◡𝐻‘𝑤) → ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦) ↔ (𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘(◡𝐻‘𝑤))))
22		breq2 5104	. . . . . . . . . 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 3590	. . . . . . . 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 581	. . . . . . 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 653	. . . . . 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 681	. . . . 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 3181	. . 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 6509	. 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 6509	. 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 1542   ∈ wcel 2114  ∀wral 3052   class class class wbr 5100  ^◡ccnv 5631  ⟶wf 6496  –1-1-onto→wf1o 6499  ‘cfv 6500   Isom wiso 6501

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-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  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-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  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-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509

This theorem is referenced by:  isores1  7290  isofr  7298  isose  7299  isopo  7302  isoso  7304  weisoeq  7311  weisoeq2  7312  fnwelem  8083  oieu  9456  oemapwe  9615  cantnffval2  9616  wemapwe  9618  infxpenlem  9935  fpwwe2lem6  10559  fpwwe2lem8  10561  infrenegsup  12137  ltweuz  13896  fz1isolem  14396  ordthmeo  23758  relogiso  26575  erdsze2lem2  35417  fzisoeu  45659