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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 4025 | . . 3 ⊢ (𝐴 ∈ V → {𝐴} ∈ V) | |
2 | rnexg 4711 | . . 3 ⊢ ({𝐴} ∈ V → ran {𝐴} ∈ V) | |
3 | uniexg 4275 | . . 3 ⊢ (ran {𝐴} ∈ V → ∪ ran {𝐴} ∈ V) | |
4 | 1, 2, 3 | 3syl 17 | . 2 ⊢ (𝐴 ∈ V → ∪ ran {𝐴} ∈ V) |
5 | sneq 3461 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
6 | 5 | rneqd 4677 | . . . 4 ⊢ (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴}) |
7 | 6 | unieqd 3670 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ ran {𝑥} = ∪ ran {𝐴}) |
8 | df-2nd 5926 | . . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) | |
9 | 7, 8 | fvmptg 5393 | . 2 ⊢ ((𝐴 ∈ V ∧ ∪ ran {𝐴} ∈ V) → (2nd ‘𝐴) = ∪ ran {𝐴}) |
10 | 4, 9 | mpdan 413 | 1 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1290 ∈ wcel 1439 Vcvv 2620 {csn 3450 ∪ cuni 3659 ran crn 4453 ‘cfv 5028 2nd c2nd 5924 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-sbc 2842 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-mpt 3907 df-id 4129 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-iota 4993 df-fun 5030 df-fv 5036 df-2nd 5926 |
This theorem is referenced by: 2nd0 5930 op2nd 5932 elxp6 5954 |
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