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Theorem isofr2 7356
Description: A weak form of isofr 7354 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 481 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵𝑉) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
2 imassrn 6077 . . . 4 (𝐻𝑥) ⊆ ran 𝐻
3 isof1o 7335 . . . . 5 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
4 f1of 6842 . . . . 5 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴𝐵)
5 frn 6732 . . . . 5 (𝐻:𝐴𝐵 → ran 𝐻𝐵)
63, 4, 53syl 18 . . . 4 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ran 𝐻𝐵)
72, 6sstrid 3991 . . 3 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐻𝑥) ⊆ 𝐵)
8 ssexg 5325 . . 3 (((𝐻𝑥) ⊆ 𝐵𝐵𝑉) → (𝐻𝑥) ∈ V)
97, 8sylan 578 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵𝑉) → (𝐻𝑥) ∈ V)
101, 9isofrlem 7352 1 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵𝑉) → (𝑆 Fr 𝐵𝑅 Fr 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2098  Vcvv 3471  wss 3947   Fr wfr 5632  ran crn 5681  cima 5683  wf 6547  1-1-ontowf1o 6550   Isom wiso 6552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-id 5578  df-fr 5635  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-isom 6560
This theorem is referenced by: (None)
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