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Mirrors > Home > MPE Home > Th. List > intpreima | Structured version Visualization version GIF version |
Description: Preimage of an intersection. (Contributed by FL, 28-Apr-2012.) |
Ref | Expression |
---|---|
intpreima | ⊢ ((Fun 𝐹 ∧ 𝐴 ≠ ∅) → (◡𝐹 “ ∩ 𝐴) = ∩ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intiin 4796 | . . 3 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 | |
2 | 1 | imaeq2i 5709 | . 2 ⊢ (◡𝐹 “ ∩ 𝐴) = (◡𝐹 “ ∩ 𝑥 ∈ 𝐴 𝑥) |
3 | iinpreima 6599 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ≠ ∅) → (◡𝐹 “ ∩ 𝑥 ∈ 𝐴 𝑥) = ∩ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥)) | |
4 | 2, 3 | syl5eq 2873 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ≠ ∅) → (◡𝐹 “ ∩ 𝐴) = ∩ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ≠ wne 2999 ∅c0 4146 ∩ cint 4699 ∩ ciin 4743 ◡ccnv 5345 “ cima 5349 Fun wfun 6121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-int 4700 df-iin 4745 df-br 4876 df-opab 4938 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-fv 6135 |
This theorem is referenced by: subbascn 21436 |
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