Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4567 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
2 | 1 | rneqd 5801 | . . . 4 ⊢ (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴}) |
3 | 2 | unieqd 4840 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ ran {𝑥} = ∪ ran {𝐴}) |
4 | df-2nd 7679 | . . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) | |
5 | snex 5322 | . . . . 5 ⊢ {𝐴} ∈ V | |
6 | 5 | rnex 7606 | . . . 4 ⊢ ran {𝐴} ∈ V |
7 | 6 | uniex 7454 | . . 3 ⊢ ∪ ran {𝐴} ∈ V |
8 | 3, 4, 7 | fvmpt 6761 | . 2 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴}) |
9 | fvprc 6656 | . . 3 ⊢ (¬ 𝐴 ∈ V → (2nd ‘𝐴) = ∅) | |
10 | snprc 4645 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
11 | 10 | biimpi 217 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
12 | 11 | rneqd 5801 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → ran {𝐴} = ran ∅) |
13 | rn0 5789 | . . . . . 6 ⊢ ran ∅ = ∅ | |
14 | 12, 13 | syl6eq 2869 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ran {𝐴} = ∅) |
15 | 14 | unieqd 4840 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ ran {𝐴} = ∪ ∅) |
16 | uni0 4857 | . . . 4 ⊢ ∪ ∅ = ∅ | |
17 | 15, 16 | syl6eq 2869 | . . 3 ⊢ (¬ 𝐴 ∈ V → ∪ ran {𝐴} = ∅) |
18 | 9, 17 | eqtr4d 2856 | . 2 ⊢ (¬ 𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴}) |
19 | 8, 18 | pm2.61i 183 | 1 ⊢ (2nd ‘𝐴) = ∪ ran {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∅c0 4288 {csn 4557 ∪ cuni 4830 ran crn 5549 ‘cfv 6348 2nd c2nd 7677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fv 6356 df-2nd 7679 |
This theorem is referenced by: 2ndnpr 7683 2nd0 7685 op2nd 7687 2nd2val 7707 elxp6 7712 |
Copyright terms: Public domain | W3C validator |