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Theorem isofr2 6494
Description: A weak form of isofr 6492 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 472 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵𝑉) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
2 imassrn 5396 . . . 4 (𝐻𝑥) ⊆ ran 𝐻
3 isof1o 6473 . . . . 5 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
4 f1of 6050 . . . . 5 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴𝐵)
5 frn 5966 . . . . 5 (𝐻:𝐴𝐵 → ran 𝐻𝐵)
63, 4, 53syl 18 . . . 4 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ran 𝐻𝐵)
72, 6syl5ss 3579 . . 3 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐻𝑥) ⊆ 𝐵)
8 ssexg 4732 . . 3 (((𝐻𝑥) ⊆ 𝐵𝐵𝑉) → (𝐻𝑥) ∈ V)
97, 8sylan 487 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵𝑉) → (𝐻𝑥) ∈ V)
101, 9isofrlem 6490 1 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵𝑉) → (𝑆 Fr 𝐵𝑅 Fr 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 1977  Vcvv 3173  wss 3540   Fr wfr 4994  ran crn 5039  cima 5041  wf 5800  1-1-ontowf1o 5803   Isom wiso 5805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-fr 4997  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813
This theorem is referenced by: (None)
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