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Mirrors > Home > MPE Home > Th. List > 2ndval | Structured version Visualization version 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 |
---|---|
2ndval | ⊢ (2nd ‘𝐴) = ∪ ran {𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4597 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
2 | 1 | rneqd 5894 | . . . 4 ⊢ (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴}) |
3 | 2 | unieqd 4880 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ ran {𝑥} = ∪ ran {𝐴}) |
4 | df-2nd 7923 | . . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) | |
5 | snex 5389 | . . . . 5 ⊢ {𝐴} ∈ V | |
6 | 5 | rnex 7850 | . . . 4 ⊢ ran {𝐴} ∈ V |
7 | 6 | uniex 7679 | . . 3 ⊢ ∪ ran {𝐴} ∈ V |
8 | 3, 4, 7 | fvmpt 6949 | . 2 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴}) |
9 | fvprc 6835 | . . 3 ⊢ (¬ 𝐴 ∈ V → (2nd ‘𝐴) = ∅) | |
10 | snprc 4679 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
11 | 10 | biimpi 215 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
12 | 11 | rneqd 5894 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → ran {𝐴} = ran ∅) |
13 | rn0 5882 | . . . . . 6 ⊢ ran ∅ = ∅ | |
14 | 12, 13 | eqtrdi 2793 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ran {𝐴} = ∅) |
15 | 14 | unieqd 4880 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ ran {𝐴} = ∪ ∅) |
16 | uni0 4897 | . . . 4 ⊢ ∪ ∅ = ∅ | |
17 | 15, 16 | eqtrdi 2793 | . . 3 ⊢ (¬ 𝐴 ∈ V → ∪ ran {𝐴} = ∅) |
18 | 9, 17 | eqtr4d 2780 | . 2 ⊢ (¬ 𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴}) |
19 | 8, 18 | pm2.61i 182 | 1 ⊢ (2nd ‘𝐴) = ∪ ran {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 Vcvv 3446 ∅c0 4283 {csn 4587 ∪ cuni 4866 ran crn 5635 ‘cfv 6497 2nd c2nd 7921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fv 6505 df-2nd 7923 |
This theorem is referenced by: 2ndnpr 7927 2nd0 7929 op2nd 7931 2nd2val 7951 elxp6 7956 |
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