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Mirrors > Home > MPE Home > Th. List > ex-1st | Structured version Visualization version GIF version |
Description: Example for df-1st 7168. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.) |
Ref | Expression |
---|---|
ex-1st | ⊢ (1st ‘〈3, 4〉) = 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3re 11094 | . . 3 ⊢ 3 ∈ ℝ | |
2 | 1 | elexi 3213 | . 2 ⊢ 3 ∈ V |
3 | 4re 11097 | . . 3 ⊢ 4 ∈ ℝ | |
4 | 3 | elexi 3213 | . 2 ⊢ 4 ∈ V |
5 | 2, 4 | op1st 7176 | 1 ⊢ (1st ‘〈3, 4〉) = 3 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 〈cop 4183 ‘cfv 5888 1st c1st 7166 ℝcr 9935 3c3 11071 4c4 11072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-i2m1 10004 ax-1ne0 10005 ax-rrecex 10008 ax-cnre 10009 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-1st 7168 df-2 11079 df-3 11080 df-4 11081 |
This theorem is referenced by: (None) |
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