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Theorem isowe2 7097
 Description: A weak form of isowe 7096 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.)
Assertion
Ref Expression
isowe2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻𝑥) ∈ V) → (𝑆 We 𝐵𝑅 We 𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑆   𝑥,𝐻

Proof of Theorem isowe2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simpl 485 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻𝑥) ∈ V) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
2 imaeq2 5919 . . . . . . 7 (𝑥 = 𝑦 → (𝐻𝑥) = (𝐻𝑦))
32eleq1d 2897 . . . . . 6 (𝑥 = 𝑦 → ((𝐻𝑥) ∈ V ↔ (𝐻𝑦) ∈ V))
43spvv 1999 . . . . 5 (∀𝑥(𝐻𝑥) ∈ V → (𝐻𝑦) ∈ V)
54adantl 484 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻𝑥) ∈ V) → (𝐻𝑦) ∈ V)
61, 5isofrlem 7087 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻𝑥) ∈ V) → (𝑆 Fr 𝐵𝑅 Fr 𝐴))
7 isosolem 7094 . . . 4 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵𝑅 Or 𝐴))
87adantr 483 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻𝑥) ∈ V) → (𝑆 Or 𝐵𝑅 Or 𝐴))
96, 8anim12d 610 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻𝑥) ∈ V) → ((𝑆 Fr 𝐵𝑆 Or 𝐵) → (𝑅 Fr 𝐴𝑅 Or 𝐴)))
10 df-we 5510 . 2 (𝑆 We 𝐵 ↔ (𝑆 Fr 𝐵𝑆 Or 𝐵))
11 df-we 5510 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Or 𝐴))
129, 10, 113imtr4g 298 1 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻𝑥) ∈ V) → (𝑆 We 𝐵𝑅 We 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1531   ∈ wcel 2110  Vcvv 3494   Or wor 5467   Fr wfr 5505   We wwe 5507   “ cima 5552   Isom wiso 6350 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-id 5454  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358 This theorem is referenced by:  fnwelem  7819  ltweuz  13323
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