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Mirrors > Home > ILE Home > Th. List > op1std | GIF version |
Description: Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
op1st.1 | ⊢ 𝐴 ∈ V |
op1st.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
op1std | ⊢ (𝐶 = 〈𝐴, 𝐵〉 → (1st ‘𝐶) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 5389 | . 2 ⊢ (𝐶 = 〈𝐴, 𝐵〉 → (1st ‘𝐶) = (1st ‘〈𝐴, 𝐵〉)) | |
2 | op1st.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | op1st.2 | . . 3 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | op1st 6012 | . 2 ⊢ (1st ‘〈𝐴, 𝐵〉) = 𝐴 |
5 | 1, 4 | syl6eq 2166 | 1 ⊢ (𝐶 = 〈𝐴, 𝐵〉 → (1st ‘𝐶) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∈ wcel 1465 Vcvv 2660 〈cop 3500 ‘cfv 5093 1st c1st 6004 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fv 5101 df-1st 6006 |
This theorem is referenced by: xp1st 6031 sbcopeq1a 6053 csbopeq1a 6054 eloprabi 6062 mpomptsx 6063 dmmpossx 6065 fmpox 6066 fmpoco 6081 df1st2 6084 xporderlem 6096 xpf1o 6706 fisumcom2 11175 txbas 12354 cnmpt1st 12384 txhmeo 12415 |
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