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Mirrors > Home > ILE Home > Th. List > fvsng | GIF version |
Description: The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.) |
Ref | Expression |
---|---|
fvsng | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1 3765 | . . . . 5 ⊢ (𝑎 = 𝐴 → 〈𝑎, 𝑏〉 = 〈𝐴, 𝑏〉) | |
2 | 1 | sneqd 3596 | . . . 4 ⊢ (𝑎 = 𝐴 → {〈𝑎, 𝑏〉} = {〈𝐴, 𝑏〉}) |
3 | id 19 | . . . 4 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴) | |
4 | 2, 3 | fveq12d 5503 | . . 3 ⊢ (𝑎 = 𝐴 → ({〈𝑎, 𝑏〉}‘𝑎) = ({〈𝐴, 𝑏〉}‘𝐴)) |
5 | 4 | eqeq1d 2179 | . 2 ⊢ (𝑎 = 𝐴 → (({〈𝑎, 𝑏〉}‘𝑎) = 𝑏 ↔ ({〈𝐴, 𝑏〉}‘𝐴) = 𝑏)) |
6 | opeq2 3766 | . . . . 5 ⊢ (𝑏 = 𝐵 → 〈𝐴, 𝑏〉 = 〈𝐴, 𝐵〉) | |
7 | 6 | sneqd 3596 | . . . 4 ⊢ (𝑏 = 𝐵 → {〈𝐴, 𝑏〉} = {〈𝐴, 𝐵〉}) |
8 | 7 | fveq1d 5498 | . . 3 ⊢ (𝑏 = 𝐵 → ({〈𝐴, 𝑏〉}‘𝐴) = ({〈𝐴, 𝐵〉}‘𝐴)) |
9 | id 19 | . . 3 ⊢ (𝑏 = 𝐵 → 𝑏 = 𝐵) | |
10 | 8, 9 | eqeq12d 2185 | . 2 ⊢ (𝑏 = 𝐵 → (({〈𝐴, 𝑏〉}‘𝐴) = 𝑏 ↔ ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵)) |
11 | vex 2733 | . . 3 ⊢ 𝑎 ∈ V | |
12 | vex 2733 | . . 3 ⊢ 𝑏 ∈ V | |
13 | 11, 12 | fvsn 5691 | . 2 ⊢ ({〈𝑎, 𝑏〉}‘𝑎) = 𝑏 |
14 | 5, 10, 13 | vtocl2g 2794 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 {csn 3583 〈cop 3586 ‘cfv 5198 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 |
This theorem is referenced by: fsnunfv 5697 fvpr1g 5702 fvpr2g 5703 tfr0dm 6301 fseq1p1m1 10050 1fv 10095 sumsnf 11372 prodsnf 11555 setsslid 12466 mgm1 12624 sgrp1 12651 mnd1 12679 mnd1id 12680 |
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