Theorem isocnv 7139

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 6673	. . . 4 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → ◡𝐻:𝐵–1-1-onto→𝐴)
2	1	adantr 484	. . 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 7088	. . . . . . . 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 7088	. . . . . . . 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 5066	. . . . . 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 6661	. . . . . . 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		ffvelrn 6902	. . . . . . . . 9 ⊢ ((◡𝐻:𝐵⟶𝐴 ∧ 𝑧 ∈ 𝐵) → (◡𝐻‘𝑧) ∈ 𝐴)
12		ffvelrn 6902	. . . . . . . . 9 ⊢ ((◡𝐻:𝐵⟶𝐴 ∧ 𝑤 ∈ 𝐵) → (◡𝐻‘𝑤) ∈ 𝐴)
13	11, 12	anim12dan 622	. . . . . . . 8 ⊢ ((◡𝐻:𝐵⟶𝐴 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((◡𝐻‘𝑧) ∈ 𝐴 ∧ (◡𝐻‘𝑤) ∈ 𝐴))
14		breq1 5056	. . . . . . . . . . 11 ⊢ (𝑥 = (◡𝐻‘𝑧) → (𝑥𝑅𝑦 ↔ (◡𝐻‘𝑧)𝑅𝑦))
15		fveq2 6717	. . . . . . . . . . . 12 ⊢ (𝑥 = (◡𝐻‘𝑧) → (𝐻‘𝑥) = (𝐻‘(◡𝐻‘𝑧)))
16	15	breq1d 5063	. . . . . . . . . . 11 ⊢ (𝑥 = (◡𝐻‘𝑧) → ((𝐻‘𝑥)𝑆(𝐻‘𝑦) ↔ (𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦)))
17	14, 16	bibi12d 349	. . . . . . . . . 10 ⊢ (𝑥 = (◡𝐻‘𝑧) → ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ ((◡𝐻‘𝑧)𝑅𝑦 ↔ (𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦))))
18		bicom 225	. . . . . . . . . 10 ⊢ (((◡𝐻‘𝑧)𝑅𝑦 ↔ (𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦)) ↔ ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦) ↔ (◡𝐻‘𝑧)𝑅𝑦))
19	17, 18	bitrdi 290	. . . . . . . . 9 ⊢ (𝑥 = (◡𝐻‘𝑧) → ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦) ↔ (◡𝐻‘𝑧)𝑅𝑦)))
20		fveq2 6717	. . . . . . . . . . 11 ⊢ (𝑦 = (◡𝐻‘𝑤) → (𝐻‘𝑦) = (𝐻‘(◡𝐻‘𝑤)))
21	20	breq2d 5065	. . . . . . . . . 10 ⊢ (𝑦 = (◡𝐻‘𝑤) → ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦) ↔ (𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘(◡𝐻‘𝑤))))
22		breq2 5057	. . . . . . . . . 10 ⊢ (𝑦 = (◡𝐻‘𝑤) → ((◡𝐻‘𝑧)𝑅𝑦 ↔ (◡𝐻‘𝑧)𝑅(◡𝐻‘𝑤)))
23	21, 22	bibi12d 349	. . . . . . . . 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 3548	. . . . . . . 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 583	. . . . . . 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 284	. . . 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 3112	. . 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 515	. 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 6389	. 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 6389	. 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 295	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110  ∀wral 3061   class class class wbr 5053  ^◡ccnv 5550  ⟶wf 6376  –1-1-onto→wf1o 6379  ‘cfv 6380   Isom wiso 6381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389

This theorem is referenced by:  isores1  7143  isofr  7151  isose  7152  isopo  7155  isoso  7157  weisoeq  7164  weisoeq2  7165  fnwelem  7898  oieu  9155  oemapwe  9309  cantnffval2  9310  wemapwe  9312  infxpenlem  9627  fpwwe2lem6  10250  fpwwe2lem8  10252  infrenegsup  11815  ltweuz  13534  fz1isolem  14027  ordthmeo  22699  relogiso  25486  erdsze2lem2  32879  fzisoeu  42512