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Mirrors > Home > MPE Home > Th. List > funfvop | Structured version Visualization version GIF version |
Description: Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 14-Oct-1996.) |
Ref | Expression |
---|---|
funfvop | ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2739 | . 2 ⊢ (𝐹‘𝐴) = (𝐹‘𝐴) | |
2 | funopfvb 6738 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) = (𝐹‘𝐴) ↔ 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹)) | |
3 | 1, 2 | mpbii 236 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 〈cop 4532 dom cdm 5535 Fun wfun 6344 ‘cfv 6350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pr 5306 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ral 3059 df-rex 3060 df-v 3402 df-sbc 3686 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-iota 6308 df-fun 6352 df-fn 6353 df-fv 6358 |
This theorem is referenced by: funfvbrb 6841 fvimacnv 6843 fnopfv 6866 fvelrn 6867 dff3 6889 fnsnb 6951 funfvima3 7022 wfrlem17 8013 tfrlem9a 8064 fundmen 8643 adj1 29881 fgreu 30597 fnsnbt 39831 |
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