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Mirrors > Home > MPE Home > Th. List > funiunfvf | Structured version Visualization version GIF version |
Description: The indexed union of a function's values is the union of its image under the index class. This version of funiunfv 7006 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.) (Revised by David Abernethy, 15-Apr-2013.) |
Ref | Expression |
---|---|
funiunfvf.1 | ⊢ Ⅎ𝑥𝐹 |
Ref | Expression |
---|---|
funiunfvf | ⊢ (Fun 𝐹 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ (𝐹 “ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funiunfvf.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
2 | nfcv 2977 | . . . 4 ⊢ Ⅎ𝑥𝑧 | |
3 | 1, 2 | nffv 6679 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
4 | nfcv 2977 | . . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) | |
5 | fveq2 6669 | . . 3 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
6 | 3, 4, 5 | cbviun 4960 | . 2 ⊢ ∪ 𝑧 ∈ 𝐴 (𝐹‘𝑧) = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) |
7 | funiunfv 7006 | . 2 ⊢ (Fun 𝐹 → ∪ 𝑧 ∈ 𝐴 (𝐹‘𝑧) = ∪ (𝐹 “ 𝐴)) | |
8 | 6, 7 | syl5eqr 2870 | 1 ⊢ (Fun 𝐹 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ (𝐹 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 Ⅎwnfc 2961 ∪ cuni 4837 ∪ ciun 4918 “ cima 5557 Fun wfun 6348 ‘cfv 6354 |
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 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 |
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 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-fv 6362 |
This theorem is referenced by: (None) |
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