Theorem isowe2 6600

Description: A weak form of isowe 6599 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 473	. . . 4 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻 “ 𝑥) ∈ V) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
2		imaeq2 5462	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐻 “ 𝑥) = (𝐻 “ 𝑦))
3	2	eleq1d 2686	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐻 “ 𝑥) ∈ V ↔ (𝐻 “ 𝑦) ∈ V))
4	3	spv 2260	. . . . 5 ⊢ (∀𝑥(𝐻 “ 𝑥) ∈ V → (𝐻 “ 𝑦) ∈ V)
5	4	adantl 482	. . . 4 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻 “ 𝑥) ∈ V) → (𝐻 “ 𝑦) ∈ V)
6	1, 5	isofrlem 6590	. . 3 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻 “ 𝑥) ∈ V) → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))
7		isosolem 6597	. . . 4 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵 → 𝑅 Or 𝐴))
8	7	adantr 481	. . 3 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻 “ 𝑥) ∈ V) → (𝑆 Or 𝐵 → 𝑅 Or 𝐴))
9	6, 8	anim12d 586	. 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 5075	. 2 ⊢ (𝑆 We 𝐵 ↔ (𝑆 Fr 𝐵 ∧ 𝑆 Or 𝐵))
11		df-we 5075	. 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))
12	9, 10, 11	3imtr4g 285	1 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻 “ 𝑥) ∈ V) → (𝑆 We 𝐵 → 𝑅 We 𝐴))

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1481   ∈ wcel 1990  Vcvv 3200   Or wor 5034   Fr wfr 5070   We wwe 5072   “ cima 5117   Isom wiso 5889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897

This theorem is referenced by:  fnwelem  7292  ltweuz  12760