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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 4580 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
2 | 1 | dmeqd 5777 | . . . 4 ⊢ (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴}) |
3 | 2 | unieqd 4855 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ dom {𝑥} = ∪ dom {𝐴}) |
4 | df-1st 7692 | . . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥}) | |
5 | snex 5335 | . . . . 5 ⊢ {𝐴} ∈ V | |
6 | 5 | dmex 7619 | . . . 4 ⊢ dom {𝐴} ∈ V |
7 | 6 | uniex 7470 | . . 3 ⊢ ∪ dom {𝐴} ∈ V |
8 | 3, 4, 7 | fvmpt 6771 | . 2 ⊢ (𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴}) |
9 | fvprc 6666 | . . 3 ⊢ (¬ 𝐴 ∈ V → (1st ‘𝐴) = ∅) | |
10 | snprc 4656 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
11 | 10 | biimpi 218 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
12 | 11 | dmeqd 5777 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → dom {𝐴} = dom ∅) |
13 | dm0 5793 | . . . . . 6 ⊢ dom ∅ = ∅ | |
14 | 12, 13 | syl6eq 2875 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → dom {𝐴} = ∅) |
15 | 14 | unieqd 4855 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ dom {𝐴} = ∪ ∅) |
16 | uni0 4869 | . . . 4 ⊢ ∪ ∅ = ∅ | |
17 | 15, 16 | syl6eq 2875 | . . 3 ⊢ (¬ 𝐴 ∈ V → ∪ dom {𝐴} = ∅) |
18 | 9, 17 | eqtr4d 2862 | . 2 ⊢ (¬ 𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴}) |
19 | 8, 18 | pm2.61i 184 | 1 ⊢ (1st ‘𝐴) = ∪ dom {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ∅c0 4294 {csn 4570 ∪ cuni 4841 dom cdm 5558 ‘cfv 6358 1st c1st 7690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-iota 6317 df-fun 6360 df-fv 6366 df-1st 7692 |
This theorem is referenced by: 1stnpr 7696 1st0 7698 op1st 7700 1st2val 7720 elxp6 7726 mpoxopxnop0 7884 |
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