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Theorem isowe2 6854
Description: A weak form of isowe 6853 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 476 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻𝑥) ∈ V) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
2 imaeq2 5702 . . . . . . 7 (𝑥 = 𝑦 → (𝐻𝑥) = (𝐻𝑦))
32eleq1d 2890 . . . . . 6 (𝑥 = 𝑦 → ((𝐻𝑥) ∈ V ↔ (𝐻𝑦) ∈ V))
43spv 2413 . . . . 5 (∀𝑥(𝐻𝑥) ∈ V → (𝐻𝑦) ∈ V)
54adantl 475 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻𝑥) ∈ V) → (𝐻𝑦) ∈ V)
61, 5isofrlem 6844 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻𝑥) ∈ V) → (𝑆 Fr 𝐵𝑅 Fr 𝐴))
7 isosolem 6851 . . . 4 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵𝑅 Or 𝐴))
87adantr 474 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻𝑥) ∈ V) → (𝑆 Or 𝐵𝑅 Or 𝐴))
96, 8anim12d 604 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻𝑥) ∈ V) → ((𝑆 Fr 𝐵𝑆 Or 𝐵) → (𝑅 Fr 𝐴𝑅 Or 𝐴)))
10 df-we 5302 . 2 (𝑆 We 𝐵 ↔ (𝑆 Fr 𝐵𝑆 Or 𝐵))
11 df-we 5302 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Or 𝐴))
129, 10, 113imtr4g 288 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 386  wal 1656  wcel 2166  Vcvv 3413   Or wor 5261   Fr wfr 5297   We wwe 5299  cima 5344   Isom wiso 6123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pr 5126
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-br 4873  df-opab 4935  df-id 5249  df-po 5262  df-so 5263  df-fr 5300  df-we 5302  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-isom 6131
This theorem is referenced by:  fnwelem  7555  ltweuz  13054
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