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Theorem isofr2 6709
Description: A weak form of isofr 6707 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 474 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵𝑉) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
2 imassrn 5587 . . . 4 (𝐻𝑥) ⊆ ran 𝐻
3 isof1o 6688 . . . . 5 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
4 f1of 6250 . . . . 5 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴𝐵)
5 frn 6166 . . . . 5 (𝐻:𝐴𝐵 → ran 𝐻𝐵)
63, 4, 53syl 18 . . . 4 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ran 𝐻𝐵)
72, 6syl5ss 3720 . . 3 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐻𝑥) ⊆ 𝐵)
8 ssexg 4912 . . 3 (((𝐻𝑥) ⊆ 𝐵𝐵𝑉) → (𝐻𝑥) ∈ V)
97, 8sylan 489 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵𝑉) → (𝐻𝑥) ∈ V)
101, 9isofrlem 6705 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 2103  Vcvv 3304  wss 3680   Fr wfr 5174  ran crn 5219  cima 5221  wf 5997  1-1-ontowf1o 6000   Isom wiso 6002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-id 5128  df-fr 5177  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-isom 6010
This theorem is referenced by: (None)
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