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Mirrors > Home > MPE Home > Th. List > Mathboxes > uniimafveqt | Structured version Visualization version GIF version |
Description: The union of the image of a subset 𝑆 of the domain of a function with elements having the same function value is the function value at one of the elements of 𝑆. (Contributed by AV, 5-Mar-2024.) |
Ref | Expression |
---|---|
uniimafveqt | ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → (∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋) → ∪ (𝐹 “ 𝑆) = (𝐹‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffun 6717 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹) | |
2 | 1 | 3ad2ant1 1134 | . . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → Fun 𝐹) |
3 | 2 | adantr 482 | . . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋)) → Fun 𝐹) |
4 | funiunfv 7242 | . . . 4 ⊢ (Fun 𝐹 → ∪ 𝑦 ∈ 𝑆 (𝐹‘𝑦) = ∪ (𝐹 “ 𝑆)) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋)) → ∪ 𝑦 ∈ 𝑆 (𝐹‘𝑦) = ∪ (𝐹 “ 𝑆)) |
6 | simp3 1139 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆) | |
7 | fveqeq2 6897 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) = (𝐹‘𝑋) ↔ (𝐹‘𝑦) = (𝐹‘𝑋))) | |
8 | 7 | cbvralvw 3235 | . . . . 5 ⊢ (∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋) ↔ ∀𝑦 ∈ 𝑆 (𝐹‘𝑦) = (𝐹‘𝑋)) |
9 | 8 | biimpi 215 | . . . 4 ⊢ (∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋) → ∀𝑦 ∈ 𝑆 (𝐹‘𝑦) = (𝐹‘𝑋)) |
10 | fveq2 6888 | . . . . 5 ⊢ (𝑦 = 𝑋 → (𝐹‘𝑦) = (𝐹‘𝑋)) | |
11 | 10 | iuneqconst 5007 | . . . 4 ⊢ ((𝑋 ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (𝐹‘𝑦) = (𝐹‘𝑋)) → ∪ 𝑦 ∈ 𝑆 (𝐹‘𝑦) = (𝐹‘𝑋)) |
12 | 6, 9, 11 | syl2an 597 | . . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋)) → ∪ 𝑦 ∈ 𝑆 (𝐹‘𝑦) = (𝐹‘𝑋)) |
13 | 5, 12 | eqtr3d 2775 | . 2 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋)) → ∪ (𝐹 “ 𝑆) = (𝐹‘𝑋)) |
14 | 13 | ex 414 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → (∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋) → ∪ (𝐹 “ 𝑆) = (𝐹‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ⊆ wss 3947 ∪ cuni 4907 ∪ ciun 4996 “ cima 5678 Fun wfun 6534 ⟶wf 6536 ‘cfv 6540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fv 6548 |
This theorem is referenced by: uniimaprimaeqfv 45985 |
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