Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4738 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → dom 𝐴 ⊆ dom 𝐵) | |
2 | dmss 4738 | . . 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 3112 | . 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
5 | eqss 3112 | . 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 1331 ⊆ wss 3071 dom cdm 4539 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-dm 4549 |
This theorem is referenced by: dmeqi 4740 dmeqd 4741 xpid11 4762 sqxpeq0 4962 fneq1 5211 eqfnfv2 5519 offval 5989 ofrfval 5990 offval3 6032 smoeq 6187 tfrlemi14d 6230 tfr1onlemres 6246 tfrcllemres 6259 rdgivallem 6278 rdgon 6283 rdg0 6284 frec0g 6294 freccllem 6299 frecfcllem 6301 frecsuclem 6303 frecsuc 6304 ereq1 6436 fundmeng 6701 acfun 7063 ccfunen 7079 ennnfonelemj0 11914 ennnfonelemg 11916 ennnfonelemp1 11919 ennnfonelemom 11921 ennnfonelemnn0 11935 blfvalps 12554 reldvg 12817 |
Copyright terms: Public domain | W3C validator |