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Theorem oacomf1olem 7641
Description: Lemma for oacomf1o 7642. (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 7640 . . . . . . 7 (𝐵 ∈ On → (𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)):On–1-1-onto→(On ∖ 𝐵))
21adantl 482 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)):On–1-1-onto→(On ∖ 𝐵))
3 f1of1 6134 . . . . . 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 6987 . . . . . 6 (𝐴 ∈ On → 𝐴 ⊆ On)
65adantr 481 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ On)
7 f1ssres 6106 . . . . 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 693 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)) ↾ 𝐴):𝐴1-1→(On ∖ 𝐵))
96resmptd 5450 . . . . . 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 2673 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ On ↦ (𝐵 +𝑜 𝑥)) ↾ 𝐴) = 𝐹)
12 f1eq1 6094 . . . . 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 6146 . . 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 6099 . . . 4 (𝐹:𝐴1-1→(On ∖ 𝐵) → 𝐹:𝐴⟶(On ∖ 𝐵))
18 frn 6051 . . . 4 (𝐹:𝐴⟶(On ∖ 𝐵) → ran 𝐹 ⊆ (On ∖ 𝐵))
1914, 17, 183syl 18 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ran 𝐹 ⊆ (On ∖ 𝐵))
2019difss2d 3738 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ran 𝐹 ⊆ On)
21 reldisj 4018 . . . 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 554 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 384   = wceq 1482  wcel 1989  cdif 3569  cin 3571  wss 3572  c0 3913  cmpt 4727  ran crn 5113  cres 5114  Oncon0 5721  wf 5882  1-1wf1 5883  1-1-ontowf1o 5885  (class class class)co 6647   +𝑜 coa 7554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-oadd 7561
This theorem is referenced by:  oacomf1o  7642  onacda  9016
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