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Mirrors > Home > MPE Home > Th. List > 1stval | Structured version Visualization version GIF version |
Description: The value of the function that extracts the first member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
1stval | ⊢ (1st ‘𝐴) = ∪ dom {𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4548 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
2 | 1 | dmeqd 5771 | . . . 4 ⊢ (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴}) |
3 | 2 | unieqd 4830 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ dom {𝑥} = ∪ dom {𝐴}) |
4 | df-1st 7758 | . . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥}) | |
5 | snex 5321 | . . . . 5 ⊢ {𝐴} ∈ V | |
6 | 5 | dmex 7686 | . . . 4 ⊢ dom {𝐴} ∈ V |
7 | 6 | uniex 7526 | . . 3 ⊢ ∪ dom {𝐴} ∈ V |
8 | 3, 4, 7 | fvmpt 6815 | . 2 ⊢ (𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴}) |
9 | fvprc 6706 | . . 3 ⊢ (¬ 𝐴 ∈ V → (1st ‘𝐴) = ∅) | |
10 | snprc 4630 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
11 | 10 | biimpi 219 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
12 | 11 | dmeqd 5771 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → dom {𝐴} = dom ∅) |
13 | dm0 5786 | . . . . . 6 ⊢ dom ∅ = ∅ | |
14 | 12, 13 | eqtrdi 2794 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → dom {𝐴} = ∅) |
15 | 14 | unieqd 4830 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ dom {𝐴} = ∪ ∅) |
16 | uni0 4846 | . . . 4 ⊢ ∪ ∅ = ∅ | |
17 | 15, 16 | eqtrdi 2794 | . . 3 ⊢ (¬ 𝐴 ∈ V → ∪ dom {𝐴} = ∅) |
18 | 9, 17 | eqtr4d 2780 | . 2 ⊢ (¬ 𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴}) |
19 | 8, 18 | pm2.61i 185 | 1 ⊢ (1st ‘𝐴) = ∪ dom {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1543 ∈ wcel 2110 Vcvv 3405 ∅c0 4234 {csn 4538 ∪ cuni 4816 dom cdm 5548 ‘cfv 6377 1st c1st 7756 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pr 5319 ax-un 7520 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-ral 3063 df-rex 3064 df-rab 3067 df-v 3407 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-nul 4235 df-if 4437 df-sn 4539 df-pr 4541 df-op 4545 df-uni 4817 df-br 5051 df-opab 5113 df-mpt 5133 df-id 5452 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6335 df-fun 6379 df-fv 6385 df-1st 7758 |
This theorem is referenced by: 1stnpr 7762 1st0 7764 op1st 7766 1st2val 7786 elxp6 7792 mpoxopxnop0 7954 |
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