Theorem isofr2 7348

Description: A weak form of isofr 7346 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.)

Assertion

Ref	Expression
isofr2	⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵 ∈ 𝑉) → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))

Proof of Theorem isofr2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 481	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵 ∈ 𝑉) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
2		imassrn 6072	. . . 4 ⊢ (𝐻 “ 𝑥) ⊆ ran 𝐻
3		isof1o 7327	. . . . 5 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
4		f1of 6835	. . . . 5 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
5		frn 6727	. . . . 5 ⊢ (𝐻:𝐴⟶𝐵 → ran 𝐻 ⊆ 𝐵)
6	3, 4, 5	3syl 18	. . . 4 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ran 𝐻 ⊆ 𝐵)
7	2, 6	sstrid 3990	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐻 “ 𝑥) ⊆ 𝐵)
8		ssexg 5320	. . 3 ⊢ (((𝐻 “ 𝑥) ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → (𝐻 “ 𝑥) ∈ V)
9	7, 8	sylan 578	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵 ∈ 𝑉) → (𝐻 “ 𝑥) ∈ V)
10	1, 9	isofrlem 7344	1 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵 ∈ 𝑉) → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2099  Vcvv 3462   ⊆ wss 3946   Fr wfr 5626  ran crn 5675   “ cima 5677  ⟶wf 6542  –1-1-onto→wf1o 6545   Isom wiso 6547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-id 5572  df-fr 5629  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-isom 6555

This theorem is referenced by: (None)