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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 3982 | . . 3 ⊢ (𝐴 ∈ V → {𝐴} ∈ V) | |
2 | rnexg 4654 | . . 3 ⊢ ({𝐴} ∈ V → ran {𝐴} ∈ V) | |
3 | uniexg 4228 | . . 3 ⊢ (ran {𝐴} ∈ V → ∪ ran {𝐴} ∈ V) | |
4 | 1, 2, 3 | 3syl 17 | . 2 ⊢ (𝐴 ∈ V → ∪ ran {𝐴} ∈ V) |
5 | sneq 3433 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
6 | 5 | rneqd 4620 | . . . 4 ⊢ (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴}) |
7 | 6 | unieqd 3638 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ ran {𝑥} = ∪ ran {𝐴}) |
8 | df-2nd 5845 | . . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) | |
9 | 7, 8 | fvmptg 5323 | . 2 ⊢ ((𝐴 ∈ V ∧ ∪ ran {𝐴} ∈ V) → (2nd ‘𝐴) = ∪ ran {𝐴}) |
10 | 4, 9 | mpdan 412 | 1 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 ∈ wcel 1434 Vcvv 2612 {csn 3422 ∪ cuni 3627 ran crn 4400 ‘cfv 4967 2nd c2nd 5843 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 3999 ax-un 4223 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-v 2614 df-sbc 2827 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-opab 3866 df-mpt 3867 df-id 4083 df-xp 4405 df-rel 4406 df-cnv 4407 df-co 4408 df-dm 4409 df-rn 4410 df-iota 4932 df-fun 4969 df-fv 4975 df-2nd 5845 |
This theorem is referenced by: 2nd0 5849 op2nd 5851 elxp6 5873 |
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