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Mirrors > Home > ILE Home > Th. List > dmeq | GIF version |
Description: Equality theorem for domain. (Contributed by NM, 11-Aug-1994.) |
Ref | Expression |
---|---|
dmeq | ⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmss 4797 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → dom 𝐴 ⊆ dom 𝐵) | |
2 | dmss 4797 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → dom 𝐵 ⊆ dom 𝐴) | |
3 | 1, 2 | anim12i 336 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) → (dom 𝐴 ⊆ dom 𝐵 ∧ dom 𝐵 ⊆ dom 𝐴)) |
4 | eqss 3152 | . 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
5 | eqss 3152 | . 2 ⊢ (dom 𝐴 = dom 𝐵 ↔ (dom 𝐴 ⊆ dom 𝐵 ∧ dom 𝐵 ⊆ dom 𝐴)) | |
6 | 3, 4, 5 | 3imtr4i 200 | 1 ⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ⊆ wss 3111 dom cdm 4598 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-dm 4608 |
This theorem is referenced by: dmeqi 4799 dmeqd 4800 xpid11 4821 sqxpeq0 5021 fneq1 5270 eqfnfv2 5578 offval 6051 ofrfval 6052 offval3 6094 smoeq 6249 tfrlemi14d 6292 tfr1onlemres 6308 tfrcllemres 6321 rdgivallem 6340 rdgon 6345 rdg0 6346 frec0g 6356 freccllem 6361 frecfcllem 6363 frecsuclem 6365 frecsuc 6366 ereq1 6499 fundmeng 6764 acfun 7154 ccfunen 7196 ennnfonelemj0 12271 ennnfonelemg 12273 ennnfonelemp1 12276 ennnfonelemom 12278 ennnfonelemnn0 12292 blfvalps 12926 reldvg 13189 |
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