Theorem fodjuomni 7104

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 5480	. . . . . . 7 ⊢ (𝑏 = 𝑧 → (inl‘𝑏) = (inl‘𝑧))
4	3	eqeq2d 2176	. . . . . 6 ⊢ (𝑏 = 𝑧 → ((𝐹‘𝑎) = (inl‘𝑏) ↔ (𝐹‘𝑎) = (inl‘𝑧)))
5	4	cbvrexv 2690	. . . . 5 ⊢ (∃𝑏 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑏) ↔ ∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧))
6		ifbi 3535	. . . . 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 4063	. . 3 ⊢ (𝑎 ∈ 𝑂 ↦ if(∃𝑏 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑏), ∅, 1o)) = (𝑎 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧), ∅, 1o))
9		fveq2 5480	. . . . . . 7 ⊢ (𝑎 = 𝑦 → (𝐹‘𝑎) = (𝐹‘𝑦))
10	9	eqeq1d 2173	. . . . . 6 ⊢ (𝑎 = 𝑦 → ((𝐹‘𝑎) = (inl‘𝑧) ↔ (𝐹‘𝑦) = (inl‘𝑧)))
11	10	rexbidv 2465	. . . . 5 ⊢ (𝑎 = 𝑦 → (∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧) ↔ ∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧)))
12	11	ifbid 3536	. . . 4 ⊢ (𝑎 = 𝑦 → if(∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧), ∅, 1o) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o))
13	12	cbvmptv 4072	. . 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 2185	. 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 7103	1 ⊢ (𝜑 → (∃𝑥 𝑥 ∈ 𝐴 ∨ 𝐴 = ∅))

Syntax hints:   → wi 4   ↔ wb 104   ∨ wo 698   = wceq 1342  ∃wex 1479   ∈ wcel 2135  ∃wrex 2443  ∅c0 3404  ifcif 3515   ↦ cmpt 4037  –onto→wfo 5180  ‘cfv 5182  1_oc1o 6368   ⊔ cdju 6993  inlcinl 7001  Omnicomni 7089

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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-if 3516  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-id 4265  df-iord 4338  df-on 4340  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-1o 6375  df-2o 6376  df-map 6607  df-dju 6994  df-inl 7003  df-inr 7004  df-omni 7090

This theorem is referenced by:  ctssexmid  7105  exmidunben  12296  exmidsbthrlem  13735  sbthomlem  13738