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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 4484 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
2 | 1 | dmeqd 5663 | . . . 4 ⊢ (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴}) |
3 | 2 | unieqd 4757 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ dom {𝑥} = ∪ dom {𝐴}) |
4 | df-1st 7548 | . . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥}) | |
5 | snex 5226 | . . . . 5 ⊢ {𝐴} ∈ V | |
6 | 5 | dmex 7475 | . . . 4 ⊢ dom {𝐴} ∈ V |
7 | 6 | uniex 7326 | . . 3 ⊢ ∪ dom {𝐴} ∈ V |
8 | 3, 4, 7 | fvmpt 6638 | . 2 ⊢ (𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴}) |
9 | fvprc 6534 | . . 3 ⊢ (¬ 𝐴 ∈ V → (1st ‘𝐴) = ∅) | |
10 | snprc 4562 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
11 | 10 | biimpi 217 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
12 | 11 | dmeqd 5663 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → dom {𝐴} = dom ∅) |
13 | dm0 5679 | . . . . . 6 ⊢ dom ∅ = ∅ | |
14 | 12, 13 | syl6eq 2846 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → dom {𝐴} = ∅) |
15 | 14 | unieqd 4757 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ dom {𝐴} = ∪ ∅) |
16 | uni0 4774 | . . . 4 ⊢ ∪ ∅ = ∅ | |
17 | 15, 16 | syl6eq 2846 | . . 3 ⊢ (¬ 𝐴 ∈ V → ∪ dom {𝐴} = ∅) |
18 | 9, 17 | eqtr4d 2833 | . 2 ⊢ (¬ 𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴}) |
19 | 8, 18 | pm2.61i 183 | 1 ⊢ (1st ‘𝐴) = ∪ dom {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1522 ∈ wcel 2080 Vcvv 3436 ∅c0 4213 {csn 4474 ∪ cuni 4747 dom cdm 5446 ‘cfv 6228 1st c1st 7546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ral 3109 df-rex 3110 df-rab 3113 df-v 3438 df-sbc 3708 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-nul 4214 df-if 4384 df-sn 4475 df-pr 4477 df-op 4481 df-uni 4748 df-br 4965 df-opab 5027 df-mpt 5044 df-id 5351 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-iota 6192 df-fun 6230 df-fv 6236 df-1st 7548 |
This theorem is referenced by: 1stnpr 7552 1st0 7554 op1st 7556 1st2val 7576 elxp6 7582 mpoxopxnop0 7735 |
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