Theorem onprcf1of1orad 36053

Description: If 𝐹 is a bijection from the ordinals to the universe and 𝐻 is a bijection from the ordinals to the proper class 𝐵, then 𝐼 maps 𝐴 one-to-one into 𝐵. This is the ZFC version of (5 → 10) in https://tinyurl.com/hamkins-gblac in the case where one of the classes is a proper class. See onprcf1of1orbd 36054 for the case where neither class is a proper class, and see onprcf1of1ord 36055 for a theorem that covers both cases. Note that in NBG set theory the first two hypotheses would be replaced by something like (𝜑 → ∀𝑋(¬ 𝑋 ∈ V → ∃𝐹𝐹:On⟶1-1onto𝑋)). However, we cannot quantify over classes, so we split this into the case where 𝑋 is the universe and the case where 𝑋 is an arbitrary proper class 𝐵 which together are sufficient for this proof. (Contributed by BTernaryTau, 17-Sep-2026.)

Hypotheses

Ref	Expression
onprcf1of1orad.1	⊢ (𝜑 → 𝐹:On⟶1-1ontoV)
onprcf1of1orad.2	⊢ (𝜑 → (¬ 𝐵 ∈ V → 𝐻:On⟶1-1onto𝐵))
onprcf1of1orad.3	⊢ (𝜑 → ¬ 𝐵 ∈ V)
onprcf1of1orad.4	⊢ 𝐼 = (𝐻 ∘ (◡𝐹 ↾ 𝐴))

Assertion

Ref	Expression
onprcf1of1orad	⊢ (𝜑 → 𝐼:𝐴⟶1-1𝐵)

Proof of Theorem onprcf1of1orad

Step	Hyp	Ref	Expression
1		onprcf1of1orad.3	. . . . 5 ⊢ (𝜑 → ¬ 𝐵 ∈ V)
2		onprcf1of1orad.2	. . . . 5 ⊢ (𝜑 → (¬ 𝐵 ∈ V → 𝐻:On⟶1-1onto𝐵))
3	1, 2	mpd 16	. . . 4 ⊢ (𝜑 → 𝐻:On⟶1-1onto𝐵)
4		f1of1 6845	. . . 4 ⊢ (𝐻:On⟶1-1onto𝐵 → 𝐻:On⟶1-1𝐵)
5	3, 4	syl 18	. . 3 ⊢ (𝜑 → 𝐻:On⟶1-1𝐵)
6		onprcf1of1orad.1	. . . 4 ⊢ (𝜑 → 𝐹:On⟶1-1ontoV)
7		f1ocnv 6859	. . . 4 ⊢ (𝐹:On⟶1-1ontoV → ◡𝐹:V⟶1-1ontoOn)
8		f1of1 6845	. . . . 5 ⊢ (◡𝐹:V⟶1-1ontoOn → ◡𝐹:V⟶1-1On)
9		ssv 3937	. . . . 5 ⊢ 𝐴 ⊆ V
10		f1ssres 6809	. . . . 5 ⊢ ((◡𝐹:V⟶1-1On ∧ 𝐴 ⊆ V) → (◡𝐹 ↾ 𝐴):𝐴⟶1-1On)
11	8, 9, 10	sylancl 598	. . . 4 ⊢ (◡𝐹:V⟶1-1ontoOn → (◡𝐹 ↾ 𝐴):𝐴⟶1-1On)
12	6, 7, 11	3syl 19	. . 3 ⊢ (𝜑 → (◡𝐹 ↾ 𝐴):𝐴⟶1-1On)
13		f1co 6813	. . 3 ⊢ ((𝐻:On⟶1-1𝐵 ∧ (◡𝐹 ↾ 𝐴):𝐴⟶1-1On) → (𝐻 ∘ (◡𝐹 ↾ 𝐴)):𝐴⟶1-1𝐵)
14	5, 12, 13	syl2anc 596	. 2 ⊢ (𝜑 → (𝐻 ∘ (◡𝐹 ↾ 𝐴)):𝐴⟶1-1𝐵)
15		onprcf1of1orad.4	. . 3 ⊢ 𝐼 = (𝐻 ∘ (◡𝐹 ↾ 𝐴))
16		f1eq1 6795	. . 3 ⊢ (𝐼 = (𝐻 ∘ (◡𝐹 ↾ 𝐴)) → (𝐼:𝐴⟶1-1𝐵 ↔ (𝐻 ∘ (◡𝐹 ↾ 𝐴)):𝐴⟶1-1𝐵))
17	15, 16	ax-mp 5	. 2 ⊢ (𝐼:𝐴⟶1-1𝐵 ↔ (𝐻 ∘ (◡𝐹 ↾ 𝐴)):𝐴⟶1-1𝐵)
18	14, 17	sylibr 237	1 ⊢ (𝜑 → 𝐼:𝐴⟶1-1𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900  ^◡ccnv 5654   ↾ cres 5657   ∘ ccom 5659  Oncon0 6380  ⟶1-1wf1 6555  ⟶1-1ontowf1o 6557

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565

This theorem is used by:  onprcf1of1ord  36055