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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 7264 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 2899 | . . . 4 ⊢ Ⅎ𝑥𝑧 | |
3 | 1, 2 | nffv 6912 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
4 | nfcv 2899 | . . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) | |
5 | fveq2 6902 | . . 3 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
6 | 3, 4, 5 | cbviun 5043 | . 2 ⊢ ∪ 𝑧 ∈ 𝐴 (𝐹‘𝑧) = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) |
7 | funiunfv 7264 | . 2 ⊢ (Fun 𝐹 → ∪ 𝑧 ∈ 𝐴 (𝐹‘𝑧) = ∪ (𝐹 “ 𝐴)) | |
8 | 6, 7 | eqtr3id 2782 | 1 ⊢ (Fun 𝐹 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ (𝐹 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 Ⅎwnfc 2879 ∪ cuni 4912 ∪ ciun 5000 “ cima 5685 Fun wfun 6547 ‘cfv 6553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-fv 6561 |
This theorem is referenced by: (None) |
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