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Theorem isowe2 6554
Description: A weak form of isowe 6553 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 473 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻𝑥) ∈ V) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
2 imaeq2 5421 . . . . . . 7 (𝑥 = 𝑦 → (𝐻𝑥) = (𝐻𝑦))
32eleq1d 2683 . . . . . 6 (𝑥 = 𝑦 → ((𝐻𝑥) ∈ V ↔ (𝐻𝑦) ∈ V))
43spv 2259 . . . . 5 (∀𝑥(𝐻𝑥) ∈ V → (𝐻𝑦) ∈ V)
54adantl 482 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻𝑥) ∈ V) → (𝐻𝑦) ∈ V)
61, 5isofrlem 6544 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻𝑥) ∈ V) → (𝑆 Fr 𝐵𝑅 Fr 𝐴))
7 isosolem 6551 . . . 4 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵𝑅 Or 𝐴))
87adantr 481 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻𝑥) ∈ V) → (𝑆 Or 𝐵𝑅 Or 𝐴))
96, 8anim12d 585 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻𝑥) ∈ V) → ((𝑆 Fr 𝐵𝑆 Or 𝐵) → (𝑅 Fr 𝐴𝑅 Or 𝐴)))
10 df-we 5035 . 2 (𝑆 We 𝐵 ↔ (𝑆 Fr 𝐵𝑆 Or 𝐵))
11 df-we 5035 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Or 𝐴))
129, 10, 113imtr4g 285 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 384  wal 1478  wcel 1987  Vcvv 3186   Or wor 4994   Fr wfr 5030   We wwe 5032  cima 5077   Isom wiso 5848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856
This theorem is referenced by:  fnwelem  7237  ltweuz  12700
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