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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 4135 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
2 | 1 | rneqd 5274 | . . . 4 ⊢ (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴}) |
3 | 2 | unieqd 4382 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ ran {𝑥} = ∪ ran {𝐴}) |
4 | df-2nd 7060 | . . 3 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) | |
5 | snex 4835 | . . . . 5 ⊢ {𝐴} ∈ V | |
6 | 5 | rnex 6992 | . . . 4 ⊢ ran {𝐴} ∈ V |
7 | 6 | uniex 6851 | . . 3 ⊢ ∪ ran {𝐴} ∈ V |
8 | 3, 4, 7 | fvmpt 6191 | . 2 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴}) |
9 | fvprc 6097 | . . 3 ⊢ (¬ 𝐴 ∈ V → (2nd ‘𝐴) = ∅) | |
10 | snprc 4197 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
11 | 10 | biimpi 205 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
12 | 11 | rneqd 5274 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → ran {𝐴} = ran ∅) |
13 | rn0 5298 | . . . . . 6 ⊢ ran ∅ = ∅ | |
14 | 12, 13 | syl6eq 2660 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ran {𝐴} = ∅) |
15 | 14 | unieqd 4382 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ ran {𝐴} = ∪ ∅) |
16 | uni0 4401 | . . . 4 ⊢ ∪ ∅ = ∅ | |
17 | 15, 16 | syl6eq 2660 | . . 3 ⊢ (¬ 𝐴 ∈ V → ∪ ran {𝐴} = ∅) |
18 | 9, 17 | eqtr4d 2647 | . 2 ⊢ (¬ 𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴}) |
19 | 8, 18 | pm2.61i 175 | 1 ⊢ (2nd ‘𝐴) = ∪ ran {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 {csn 4125 ∪ cuni 4372 ran crn 5039 ‘cfv 5804 2nd c2nd 7058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-2nd 7060 |
This theorem is referenced by: 2ndnpr 7064 2nd0 7066 op2nd 7068 2nd2val 7086 elxp6 7091 |
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