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 4535 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
2 | 1 | rneqd 5772 | . . . 4 ⊢ (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴}) |
3 | 2 | unieqd 4814 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ ran {𝑥} = ∪ ran {𝐴}) |
4 | df-2nd 7672 | . . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) | |
5 | snex 5297 | . . . . 5 ⊢ {𝐴} ∈ V | |
6 | 5 | rnex 7599 | . . . 4 ⊢ ran {𝐴} ∈ V |
7 | 6 | uniex 7447 | . . 3 ⊢ ∪ ran {𝐴} ∈ V |
8 | 3, 4, 7 | fvmpt 6745 | . 2 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴}) |
9 | fvprc 6638 | . . 3 ⊢ (¬ 𝐴 ∈ V → (2nd ‘𝐴) = ∅) | |
10 | snprc 4613 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
11 | 10 | biimpi 219 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
12 | 11 | rneqd 5772 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → ran {𝐴} = ran ∅) |
13 | rn0 5760 | . . . . . 6 ⊢ ran ∅ = ∅ | |
14 | 12, 13 | eqtrdi 2849 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ran {𝐴} = ∅) |
15 | 14 | unieqd 4814 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ ran {𝐴} = ∪ ∅) |
16 | uni0 4828 | . . . 4 ⊢ ∪ ∅ = ∅ | |
17 | 15, 16 | eqtrdi 2849 | . . 3 ⊢ (¬ 𝐴 ∈ V → ∪ ran {𝐴} = ∅) |
18 | 9, 17 | eqtr4d 2836 | . 2 ⊢ (¬ 𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴}) |
19 | 8, 18 | pm2.61i 185 | 1 ⊢ (2nd ‘𝐴) = ∪ ran {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 {csn 4525 ∪ cuni 4800 ran crn 5520 ‘cfv 6324 2nd c2nd 7670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fv 6332 df-2nd 7672 |
This theorem is referenced by: 2ndnpr 7676 2nd0 7678 op2nd 7680 2nd2val 7700 elxp6 7705 |
Copyright terms: Public domain | W3C validator |