Theorem isofr2 6967

Description: A weak form of isofr 6965 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 483	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵 ∈ 𝑉) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
2		imassrn 5824	. . . 4 ⊢ (𝐻 “ 𝑥) ⊆ ran 𝐻
3		isof1o 6946	. . . . 5 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
4		f1of 6490	. . . . 5 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
5		frn 6395	. . . . 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 3906	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐻 “ 𝑥) ⊆ 𝐵)
8		ssexg 5125	. . 3 ⊢ (((𝐻 “ 𝑥) ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → (𝐻 “ 𝑥) ∈ V)
9	7, 8	sylan 580	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵 ∈ 𝑉) → (𝐻 “ 𝑥) ∈ V)
10	1, 9	isofrlem 6963	1 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵 ∈ 𝑉) → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2083  Vcvv 3440   ⊆ wss 3865   Fr wfr 5406  ran crn 5451   “ cima 5453  ⟶wf 6228  –1-1-onto→wf1o 6231   Isom wiso 6233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-id 5355  df-fr 5409  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-isom 6241

This theorem is referenced by: (None)