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Mirrors > Home > ILE Home > Th. List > 2ndvalg | GIF version |
Description: The value of the function that extracts the second member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
2ndvalg | ⊢ (𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snexg 4202 | . . 3 ⊢ (𝐴 ∈ V → {𝐴} ∈ V) | |
2 | rnexg 4910 | . . 3 ⊢ ({𝐴} ∈ V → ran {𝐴} ∈ V) | |
3 | uniexg 4457 | . . 3 ⊢ (ran {𝐴} ∈ V → ∪ ran {𝐴} ∈ V) | |
4 | 1, 2, 3 | 3syl 17 | . 2 ⊢ (𝐴 ∈ V → ∪ ran {𝐴} ∈ V) |
5 | sneq 3618 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
6 | 5 | rneqd 4874 | . . . 4 ⊢ (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴}) |
7 | 6 | unieqd 3835 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ ran {𝑥} = ∪ ran {𝐴}) |
8 | df-2nd 6166 | . . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) | |
9 | 7, 8 | fvmptg 5613 | . 2 ⊢ ((𝐴 ∈ V ∧ ∪ ran {𝐴} ∈ V) → (2nd ‘𝐴) = ∪ ran {𝐴}) |
10 | 4, 9 | mpdan 421 | 1 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 Vcvv 2752 {csn 3607 ∪ cuni 3824 ran crn 4645 ‘cfv 5235 2nd c2nd 6164 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-iota 5196 df-fun 5237 df-fv 5243 df-2nd 6166 |
This theorem is referenced by: 2nd0 6170 op2nd 6172 elxp6 6194 |
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