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Mirrors > Home > MPE Home > Th. List > isowe2 | Structured version Visualization version GIF version |
Description: A weak form of isowe 6853 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.) |
Ref | Expression |
---|---|
isowe2 | ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻 “ 𝑥) ∈ V) → (𝑆 We 𝐵 → 𝑅 We 𝐴)) |
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)) |
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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