Theorem isofr 7283

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 7271	. . 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 7264	. . . . 5 ⊢ (◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) → ◡𝐻:𝐵–1-1-onto→𝐴)
4		f1ofun 6770	. . . . 5 ⊢ (◡𝐻:𝐵–1-1-onto→𝐴 → Fun ◡𝐻)
5		vex 3442	. . . . . 6 ⊢ 𝑥 ∈ V
6	5	funimaex 6574	. . . . 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 7281	. . 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 7264	. . . 4 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
12		f1ofun 6770	. . . 4 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → Fun 𝐻)
13	5	funimaex 6574	. . . 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 7281	. 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 2109  Vcvv 3438   Fr wfr 5573  ^◡ccnv 5622   “ cima 5626  Fun wfun 6480  –1-1-onto→wf1o 6485   Isom wiso 6487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-id 5518  df-fr 5576  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495

This theorem is referenced by:  isowe  7290  wofib  9456  isfin1-4  10300