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(𝐻 “ 𝑥) ∈ V) → (𝑆 We 𝐵 → 𝑅 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.

Step	Hyp	Ref	Expression
1		simpl 476	. . . 4 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻 “ 𝑥) ∈ V) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
2		imaeq2 5702	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐻 “ 𝑥) = (𝐻 “ 𝑦))
3	2	eleq1d 2890	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐻 “ 𝑥) ∈ V ↔ (𝐻 “ 𝑦) ∈ V))
4	3	spv 2413	. . . . 5 ⊢ (∀𝑥(𝐻 “ 𝑥) ∈ V → (𝐻 “ 𝑦) ∈ V)
5	4	adantl 475	. . . 4 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻 “ 𝑥) ∈ V) → (𝐻 “ 𝑦) ∈ V)
6	1, 5	isofrlem 6844	. . 3 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻 “ 𝑥) ∈ V) → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))
7		isosolem 6851	. . . 4 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵 → 𝑅 Or 𝐴))
8	7	adantr 474	. . 3 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻 “ 𝑥) ∈ V) → (𝑆 Or 𝐵 → 𝑅 Or 𝐴))
9	6, 8	anim12d 604	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻 “ 𝑥) ∈ V) → ((𝑆 Fr 𝐵 ∧ 𝑆 Or 𝐵) → (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)))
10		df-we 5302	. 2 ⊢ (𝑆 We 𝐵 ↔ (𝑆 Fr 𝐵 ∧ 𝑆 Or 𝐵))
11		df-we 5302	. 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))
12	9, 10, 11	3imtr4g 288	1 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻 “ 𝑥) ∈ V) → (𝑆 We 𝐵 → 𝑅 We 𝐴))

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