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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) ∧ 𝐵𝑉) → (𝑆 Fr 𝐵𝑅 Fr 𝐴))

Proof of Theorem isofr2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpl 483 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵𝑉) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
2 imassrn 5824 . . . 4 (𝐻𝑥) ⊆ ran 𝐻
3 isof1o 6946 . . . . 5 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
4 f1of 6490 . . . . 5 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴𝐵)
5 frn 6395 . . . . 5 (𝐻:𝐴𝐵 → ran 𝐻𝐵)
63, 4, 53syl 18 . . . 4 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ran 𝐻𝐵)
72, 6sstrid 3906 . . 3 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐻𝑥) ⊆ 𝐵)
8 ssexg 5125 . . 3 (((𝐻𝑥) ⊆ 𝐵𝐵𝑉) → (𝐻𝑥) ∈ V)
97, 8sylan 580 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵𝑉) → (𝐻𝑥) ∈ V)
101, 9isofrlem 6963 1 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵𝑉) → (𝑆 Fr 𝐵𝑅 Fr 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2083  Vcvv 3440  wss 3865   Fr wfr 5406  ran crn 5451  cima 5453  wf 6228  1-1-ontowf1o 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)
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