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Theorem oacomf1olem 7813
Description: Lemma for oacomf1o 7814. (Contributed by Mario Carneiro, 30-May-2015.)
Hypothesis
Ref Expression
oacomf1olem.1 𝐹 = (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥))
Assertion
Ref Expression
oacomf1olem ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹:𝐴1-1-onto→ran 𝐹 ∧ (ran 𝐹𝐵) = ∅))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem oacomf1olem
StepHypRef Expression
1 oaf1o 7812 . . . . . . 7 (𝐵 ∈ On → (𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)):On–1-1-onto→(On ∖ 𝐵))
21adantl 473 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)):On–1-1-onto→(On ∖ 𝐵))
3 f1of1 6297 . . . . . 6 ((𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)):On–1-1-onto→(On ∖ 𝐵) → (𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)):On–1-1→(On ∖ 𝐵))
42, 3syl 17 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)):On–1-1→(On ∖ 𝐵))
5 onss 7155 . . . . . 6 (𝐴 ∈ On → 𝐴 ⊆ On)
65adantr 472 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ On)
7 f1ssres 6269 . . . . 5 (((𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)):On–1-1→(On ∖ 𝐵) ∧ 𝐴 ⊆ On) → ((𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)) ↾ 𝐴):𝐴1-1→(On ∖ 𝐵))
84, 6, 7syl2anc 696 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)) ↾ 𝐴):𝐴1-1→(On ∖ 𝐵))
96resmptd 5610 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)) ↾ 𝐴) = (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)))
10 oacomf1olem.1 . . . . . 6 𝐹 = (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥))
119, 10syl6eqr 2812 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)) ↾ 𝐴) = 𝐹)
12 f1eq1 6257 . . . . 5 (((𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)) ↾ 𝐴) = 𝐹 → (((𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)) ↾ 𝐴):𝐴1-1→(On ∖ 𝐵) ↔ 𝐹:𝐴1-1→(On ∖ 𝐵)))
1311, 12syl 17 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (((𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)) ↾ 𝐴):𝐴1-1→(On ∖ 𝐵) ↔ 𝐹:𝐴1-1→(On ∖ 𝐵)))
148, 13mpbid 222 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:𝐴1-1→(On ∖ 𝐵))
15 f1f1orn 6309 . . 3 (𝐹:𝐴1-1→(On ∖ 𝐵) → 𝐹:𝐴1-1-onto→ran 𝐹)
1614, 15syl 17 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:𝐴1-1-onto→ran 𝐹)
17 f1f 6262 . . . 4 (𝐹:𝐴1-1→(On ∖ 𝐵) → 𝐹:𝐴⟶(On ∖ 𝐵))
18 frn 6214 . . . 4 (𝐹:𝐴⟶(On ∖ 𝐵) → ran 𝐹 ⊆ (On ∖ 𝐵))
1914, 17, 183syl 18 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ran 𝐹 ⊆ (On ∖ 𝐵))
2019difss2d 3883 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ran 𝐹 ⊆ On)
21 reldisj 4163 . . . 4 (ran 𝐹 ⊆ On → ((ran 𝐹𝐵) = ∅ ↔ ran 𝐹 ⊆ (On ∖ 𝐵)))
2220, 21syl 17 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((ran 𝐹𝐵) = ∅ ↔ ran 𝐹 ⊆ (On ∖ 𝐵)))
2319, 22mpbird 247 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran 𝐹𝐵) = ∅)
2416, 23jca 555 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹:𝐴1-1-onto→ran 𝐹 ∧ (ran 𝐹𝐵) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  cdif 3712  cin 3714  wss 3715  c0 4058  cmpt 4881  ran crn 5267  cres 5268  Oncon0 5884  wf 6045  1-1wf1 6046  1-1-ontowf1o 6048  (class class class)co 6813   +𝑜 coa 7726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-oadd 7733
This theorem is referenced by:  oacomf1o  7814  onacda  9211
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