Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvelima2 | Structured version Visualization version GIF version |
Description: Function value in an image. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
fvelima2 | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ (𝐹 “ 𝐶)) → ∃𝑥 ∈ (𝐴 ∩ 𝐶)(𝐹‘𝑥) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elimag 5927 | . . . 4 ⊢ (𝐵 ∈ (𝐹 “ 𝐶) → (𝐵 ∈ (𝐹 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐹𝐵)) | |
2 | 1 | ibi 269 | . . 3 ⊢ (𝐵 ∈ (𝐹 “ 𝐶) → ∃𝑥 ∈ 𝐶 𝑥𝐹𝐵) |
3 | df-rex 3144 | . . 3 ⊢ (∃𝑥 ∈ 𝐶 𝑥𝐹𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵)) | |
4 | 2, 3 | sylib 220 | . 2 ⊢ (𝐵 ∈ (𝐹 “ 𝐶) → ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵)) |
5 | fnbr 6453 | . . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥𝐹𝐵) → 𝑥 ∈ 𝐴) | |
6 | 5 | adantrl 714 | . . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵)) → 𝑥 ∈ 𝐴) |
7 | simprl 769 | . . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵)) → 𝑥 ∈ 𝐶) | |
8 | 6, 7 | elind 4170 | . . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵)) → 𝑥 ∈ (𝐴 ∩ 𝐶)) |
9 | fnfun 6447 | . . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
10 | funbrfv 6710 | . . . . . . . . . 10 ⊢ (Fun 𝐹 → (𝑥𝐹𝐵 → (𝐹‘𝑥) = 𝐵)) | |
11 | 10 | imp 409 | . . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥𝐹𝐵) → (𝐹‘𝑥) = 𝐵) |
12 | 9, 11 | sylan 582 | . . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥𝐹𝐵) → (𝐹‘𝑥) = 𝐵) |
13 | 12 | adantrl 714 | . . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵)) → (𝐹‘𝑥) = 𝐵) |
14 | 8, 13 | jca 514 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵)) → (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ (𝐹‘𝑥) = 𝐵)) |
15 | 14 | ex 415 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵) → (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ (𝐹‘𝑥) = 𝐵))) |
16 | 15 | eximdv 1914 | . . . 4 ⊢ (𝐹 Fn 𝐴 → (∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵) → ∃𝑥(𝑥 ∈ (𝐴 ∩ 𝐶) ∧ (𝐹‘𝑥) = 𝐵))) |
17 | 16 | imp 409 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵)) → ∃𝑥(𝑥 ∈ (𝐴 ∩ 𝐶) ∧ (𝐹‘𝑥) = 𝐵)) |
18 | df-rex 3144 | . . 3 ⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐶)(𝐹‘𝑥) = 𝐵 ↔ ∃𝑥(𝑥 ∈ (𝐴 ∩ 𝐶) ∧ (𝐹‘𝑥) = 𝐵)) | |
19 | 17, 18 | sylibr 236 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐹𝐵)) → ∃𝑥 ∈ (𝐴 ∩ 𝐶)(𝐹‘𝑥) = 𝐵) |
20 | 4, 19 | sylan2 594 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ (𝐹 “ 𝐶)) → ∃𝑥 ∈ (𝐴 ∩ 𝐶)(𝐹‘𝑥) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ∃wrex 3139 ∩ cin 3934 class class class wbr 5058 “ cima 5552 Fun wfun 6343 Fn wfn 6344 ‘cfv 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-fv 6357 |
This theorem is referenced by: limsupresxr 42040 liminfresxr 42041 liminfvalxr 42057 |
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