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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 4577 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
2 | 1 | dmeqd 5774 | . . . 4 ⊢ (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴}) |
3 | 2 | unieqd 4852 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ dom {𝑥} = ∪ dom {𝐴}) |
4 | df-1st 7689 | . . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥}) | |
5 | snex 5332 | . . . . 5 ⊢ {𝐴} ∈ V | |
6 | 5 | dmex 7616 | . . . 4 ⊢ dom {𝐴} ∈ V |
7 | 6 | uniex 7467 | . . 3 ⊢ ∪ dom {𝐴} ∈ V |
8 | 3, 4, 7 | fvmpt 6768 | . 2 ⊢ (𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴}) |
9 | fvprc 6663 | . . 3 ⊢ (¬ 𝐴 ∈ V → (1st ‘𝐴) = ∅) | |
10 | snprc 4653 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
11 | 10 | biimpi 218 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
12 | 11 | dmeqd 5774 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → dom {𝐴} = dom ∅) |
13 | dm0 5790 | . . . . . 6 ⊢ dom ∅ = ∅ | |
14 | 12, 13 | syl6eq 2872 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → dom {𝐴} = ∅) |
15 | 14 | unieqd 4852 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ dom {𝐴} = ∪ ∅) |
16 | uni0 4866 | . . . 4 ⊢ ∪ ∅ = ∅ | |
17 | 15, 16 | syl6eq 2872 | . . 3 ⊢ (¬ 𝐴 ∈ V → ∪ dom {𝐴} = ∅) |
18 | 9, 17 | eqtr4d 2859 | . 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 1537 ∈ wcel 2114 Vcvv 3494 ∅c0 4291 {csn 4567 ∪ cuni 4838 dom cdm 5555 ‘cfv 6355 1st c1st 7687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fv 6363 df-1st 7689 |
This theorem is referenced by: 1stnpr 7693 1st0 7695 op1st 7697 1st2val 7717 elxp6 7723 mpoxopxnop0 7881 |
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