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Mirrors > Home > MPE Home > Th. List > oacomf1olem | Structured version Visualization version GIF version |
Description: Lemma for oacomf1o 7532. (Contributed by Mario Carneiro, 30-May-2015.) |
Ref | Expression |
---|---|
oacomf1olem.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) |
Ref | Expression |
---|---|
oacomf1olem | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹:𝐴–1-1-onto→ran 𝐹 ∧ (ran 𝐹 ∩ 𝐵) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oaf1o 7530 | . . . . . . 7 ⊢ (𝐵 ∈ On → (𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)):On–1-1-onto→(On ∖ 𝐵)) | |
2 | 1 | adantl 481 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)):On–1-1-onto→(On ∖ 𝐵)) |
3 | f1of1 6049 | . . . . . 6 ⊢ ((𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)):On–1-1-onto→(On ∖ 𝐵) → (𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)):On–1-1→(On ∖ 𝐵)) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)):On–1-1→(On ∖ 𝐵)) |
5 | onss 6882 | . . . . . 6 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On) | |
6 | 5 | adantr 480 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ On) |
7 | f1ssres 6021 | . . . . 5 ⊢ (((𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)):On–1-1→(On ∖ 𝐵) ∧ 𝐴 ⊆ On) → ((𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)) ↾ 𝐴):𝐴–1-1→(On ∖ 𝐵)) | |
8 | 4, 6, 7 | syl2anc 691 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)) ↾ 𝐴):𝐴–1-1→(On ∖ 𝐵)) |
9 | 6 | resmptd 5371 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥))) |
10 | oacomf1olem.1 | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) | |
11 | 9, 10 | syl6eqr 2662 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)) ↾ 𝐴) = 𝐹) |
12 | f1eq1 6009 | . . . . 5 ⊢ (((𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)) ↾ 𝐴) = 𝐹 → (((𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)) ↾ 𝐴):𝐴–1-1→(On ∖ 𝐵) ↔ 𝐹:𝐴–1-1→(On ∖ 𝐵))) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (((𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)) ↾ 𝐴):𝐴–1-1→(On ∖ 𝐵) ↔ 𝐹:𝐴–1-1→(On ∖ 𝐵))) |
14 | 8, 13 | mpbid 221 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:𝐴–1-1→(On ∖ 𝐵)) |
15 | f1f1orn 6061 | . . 3 ⊢ (𝐹:𝐴–1-1→(On ∖ 𝐵) → 𝐹:𝐴–1-1-onto→ran 𝐹) | |
16 | 14, 15 | syl 17 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:𝐴–1-1-onto→ran 𝐹) |
17 | f1f 6014 | . . . 4 ⊢ (𝐹:𝐴–1-1→(On ∖ 𝐵) → 𝐹:𝐴⟶(On ∖ 𝐵)) | |
18 | frn 5966 | . . . 4 ⊢ (𝐹:𝐴⟶(On ∖ 𝐵) → ran 𝐹 ⊆ (On ∖ 𝐵)) | |
19 | 14, 17, 18 | 3syl 18 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ran 𝐹 ⊆ (On ∖ 𝐵)) |
20 | 19 | difss2d 3702 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ran 𝐹 ⊆ On) |
21 | reldisj 3972 | . . . 4 ⊢ (ran 𝐹 ⊆ On → ((ran 𝐹 ∩ 𝐵) = ∅ ↔ ran 𝐹 ⊆ (On ∖ 𝐵))) | |
22 | 20, 21 | syl 17 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((ran 𝐹 ∩ 𝐵) = ∅ ↔ ran 𝐹 ⊆ (On ∖ 𝐵))) |
23 | 19, 22 | mpbird 246 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran 𝐹 ∩ 𝐵) = ∅) |
24 | 16, 23 | jca 553 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹:𝐴–1-1-onto→ran 𝐹 ∧ (ran 𝐹 ∩ 𝐵) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ↦ cmpt 4643 ran crn 5039 ↾ cres 5040 Oncon0 5640 ⟶wf 5800 –1-1→wf1 5801 –1-1-onto→wf1o 5803 (class class class)co 6549 +𝑜 coa 7444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-oadd 7451 |
This theorem is referenced by: oacomf1o 7532 onacda 8902 |
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