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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 5700 | . . . 4 ⊢ ran 𝐹 = dom ◡𝐹 | |
2 | frn 6744 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
3 | 2 | adantr 480 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → ran 𝐹 ⊆ 𝐵) |
4 | 1, 3 | eqsstrrid 4045 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → dom ◡𝐹 ⊆ 𝐵) |
5 | ssdisj 4466 | . . 3 ⊢ ((dom ◡𝐹 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (dom ◡𝐹 ∩ 𝐶) = ∅) | |
6 | 4, 5 | sylancom 588 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (dom ◡𝐹 ∩ 𝐶) = ∅) |
7 | imadisj 6100 | . 2 ⊢ ((◡𝐹 “ 𝐶) = ∅ ↔ (dom ◡𝐹 ∩ 𝐶) = ∅) | |
8 | 6, 7 | sylibr 234 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (◡𝐹 “ 𝐶) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 ◡ccnv 5688 dom cdm 5689 ran crn 5690 “ cima 5692 ⟶wf 6559 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-f 6567 |
This theorem is referenced by: vdwmc2 17013 gsumval3a 19936 psrbag0 22104 mbfconstlem 25676 itg1val2 25733 ofpreima2 32683 |
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