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Mirrors > Home > MPE Home > Th. List > fimacnvdisj | Structured version Visualization version GIF version |
Description: The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.) |
Ref | Expression |
---|---|
fimacnvdisj | ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (◡𝐹 “ 𝐶) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rn 5551 | . . . 4 ⊢ ran 𝐹 = dom ◡𝐹 | |
2 | frn 6541 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
3 | 2 | adantr 484 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → ran 𝐹 ⊆ 𝐵) |
4 | 1, 3 | eqsstrrid 3940 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → dom ◡𝐹 ⊆ 𝐵) |
5 | ssdisj 4364 | . . 3 ⊢ ((dom ◡𝐹 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (dom ◡𝐹 ∩ 𝐶) = ∅) | |
6 | 4, 5 | sylancom 591 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (dom ◡𝐹 ∩ 𝐶) = ∅) |
7 | imadisj 5937 | . 2 ⊢ ((◡𝐹 “ 𝐶) = ∅ ↔ (dom ◡𝐹 ∩ 𝐶) = ∅) | |
8 | 6, 7 | sylibr 237 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (◡𝐹 “ 𝐶) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∩ cin 3856 ⊆ wss 3857 ∅c0 4227 ◡ccnv 5539 dom cdm 5540 ran crn 5541 “ cima 5543 ⟶wf 6365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-br 5044 df-opab 5106 df-xp 5546 df-cnv 5548 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-f 6373 |
This theorem is referenced by: vdwmc2 16513 gsumval3a 19260 psrbag0 20992 mbfconstlem 24496 itg1val2 24553 ofpreima2 30695 |
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