Theorem isofr 7290

Description: An isomorphism preserves well-foundedness. Proposition 6.32(1) of [TakeutiZaring] p. 33. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)

Assertion

Ref	Expression
isofr	⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐵))

Proof of Theorem isofr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isocnv 7278	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴))
2		id 22	. . . 4 ⊢ (◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) → ◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴))
3		isof1o 7271	. . . . 5 ⊢ (◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) → ◡𝐻:𝐵–1-1-onto→𝐴)
4		f1ofun 6776	. . . . 5 ⊢ (◡𝐻:𝐵–1-1-onto→𝐴 → Fun ◡𝐻)
5		vex 3434	. . . . . 6 ⊢ 𝑥 ∈ V
6	5	funimaex 6580	. . . . 5 ⊢ (Fun ◡𝐻 → (◡𝐻 “ 𝑥) ∈ V)
7	3, 4, 6	3syl 18	. . . 4 ⊢ (◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) → (◡𝐻 “ 𝑥) ∈ V)
8	2, 7	isofrlem 7288	. . 3 ⊢ (◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) → (𝑅 Fr 𝐴 → 𝑆 Fr 𝐵))
9	1, 8	syl 17	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Fr 𝐴 → 𝑆 Fr 𝐵))
10		id 22	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
11		isof1o 7271	. . . 4 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
12		f1ofun 6776	. . . 4 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → Fun 𝐻)
13	5	funimaex 6580	. . . 4 ⊢ (Fun 𝐻 → (𝐻 “ 𝑥) ∈ V)
14	11, 12, 13	3syl 18	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐻 “ 𝑥) ∈ V)
15	10, 14	isofrlem 7288	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))
16	9, 15	impbid 212	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2114  Vcvv 3430   Fr wfr 5574  ^◡ccnv 5623   “ cima 5627  Fun wfun 6486  –1-1-onto→wf1o 6491   Isom wiso 6493

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-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5370

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5519  df-fr 5577  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:  isowe  7297  wofib  9453  isfin1-4  10300