Theorem fodjuomni 7121

Description: A condition which ensures 𝐴 is either inhabited or empty. Lemma 3.2 of [PradicBrown2022], p. 4. (Contributed by Jim Kingdon, 27-Jul-2022.)

Hypotheses

Ref	Expression
fodjuomni.o	⊢ (𝜑 → 𝑂 ∈ Omni)
fodjuomni.fo	⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))

Assertion

Ref	Expression
fodjuomni	⊢ (𝜑 → (∃𝑥 𝑥 ∈ 𝐴 ∨ 𝐴 = ∅))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑂(𝑥)

Proof of Theorem fodjuomni

Dummy variables 𝑎 𝑏 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fodjuomni.o	. 2 ⊢ (𝜑 → 𝑂 ∈ Omni)
2		fodjuomni.fo	. 2 ⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))
3		fveq2 5494	. . . . . . 7 ⊢ (𝑏 = 𝑧 → (inl‘𝑏) = (inl‘𝑧))
4	3	eqeq2d 2182	. . . . . 6 ⊢ (𝑏 = 𝑧 → ((𝐹‘𝑎) = (inl‘𝑏) ↔ (𝐹‘𝑎) = (inl‘𝑧)))
5	4	cbvrexv 2697	. . . . 5 ⊢ (∃𝑏 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑏) ↔ ∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧))
6		ifbi 3545	. . . . 5 ⊢ ((∃𝑏 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑏) ↔ ∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧)) → if(∃𝑏 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑏), ∅, 1o) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧), ∅, 1o))
7	5, 6	ax-mp 5	. . . 4 ⊢ if(∃𝑏 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑏), ∅, 1o) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧), ∅, 1o)
8	7	mpteq2i 4074	. . 3 ⊢ (𝑎 ∈ 𝑂 ↦ if(∃𝑏 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑏), ∅, 1o)) = (𝑎 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧), ∅, 1o))
9		fveq2 5494	. . . . . . 7 ⊢ (𝑎 = 𝑦 → (𝐹‘𝑎) = (𝐹‘𝑦))
10	9	eqeq1d 2179	. . . . . 6 ⊢ (𝑎 = 𝑦 → ((𝐹‘𝑎) = (inl‘𝑧) ↔ (𝐹‘𝑦) = (inl‘𝑧)))
11	10	rexbidv 2471	. . . . 5 ⊢ (𝑎 = 𝑦 → (∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧) ↔ ∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧)))
12	11	ifbid 3546	. . . 4 ⊢ (𝑎 = 𝑦 → if(∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧), ∅, 1o) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o))
13	12	cbvmptv 4083	. . 3 ⊢ (𝑎 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧), ∅, 1o)) = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o))
14	8, 13	eqtri 2191	. 2 ⊢ (𝑎 ∈ 𝑂 ↦ if(∃𝑏 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑏), ∅, 1o)) = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o))
15	1, 2, 14	fodjuomnilemres 7120	1 ⊢ (𝜑 → (∃𝑥 𝑥 ∈ 𝐴 ∨ 𝐴 = ∅))

Syntax hints:   → wi 4   ↔ wb 104   ∨ wo 703   = wceq 1348  ∃wex 1485   ∈ wcel 2141  ∃wrex 2449  ∅c0 3414  ifcif 3525   ↦ cmpt 4048  –onto→wfo 5194  ‘cfv 5196  1_oc1o 6385   ⊔ cdju 7010  inlcinl 7018  Omnicomni 7106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-1o 6392  df-2o 6393  df-map 6624  df-dju 7011  df-inl 7020  df-inr 7021  df-omni 7107

This theorem is referenced by:  ctssexmid  7122  exmidunben  12368  exmidsbthrlem  14014  sbthomlem  14017