Theorem fodjum 7011

Description: Lemma for fodjuomni 7014 and fodjumkv 7027. A condition which shows that 𝐴 is inhabited. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)

Hypotheses

Ref	Expression
fodjuf.fo	⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))
fodjuf.p	⊢ 𝑃 = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o))
fodjum.z	⊢ (𝜑 → ∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅)

Assertion

Ref	Expression
fodjum	⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)

Distinct variable groups:   𝜑,𝑦,𝑧   𝑦,𝑂,𝑧   𝑧,𝐴   𝑧,𝐵   𝑧,𝐹   𝑤,𝐴,𝑥,𝑧   𝑦,𝐴,𝑤   𝑦,𝐹   𝜑,𝑤

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥,𝑦,𝑤)   𝑃(𝑥,𝑦,𝑧,𝑤)   𝐹(𝑥,𝑤)   𝑂(𝑥,𝑤)

Proof of Theorem fodjum

Step	Hyp	Ref	Expression
1		fodjum.z	. 2 ⊢ (𝜑 → ∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅)
2		1n0 6322	. . . . . . . . 9 ⊢ 1o ≠ ∅
3	2	nesymi 2352	. . . . . . . 8 ⊢ ¬ ∅ = 1o
4	3	intnan 914	. . . . . . 7 ⊢ ¬ (¬ ∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧) ∧ ∅ = 1o)
5	4	a1i 9	. . . . . 6 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → ¬ (¬ ∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧) ∧ ∅ = 1o))
6		simprr 521	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → (𝑃‘𝑤) = ∅)
7		fodjuf.p	. . . . . . . . 9 ⊢ 𝑃 = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o))
8		fveqeq2 5423	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑦) = (inl‘𝑧) ↔ (𝐹‘𝑤) = (inl‘𝑧)))
9	8	rexbidv 2436	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧) ↔ ∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧)))
10	9	ifbid 3488	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧), ∅, 1o))
11		simprl 520	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → 𝑤 ∈ 𝑂)
12		peano1 4503	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
13	12	a1i 9	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → ∅ ∈ ω)
14		1onn 6409	. . . . . . . . . . 11 ⊢ 1o ∈ ω
15	14	a1i 9	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → 1o ∈ ω)
16		fodjuf.fo	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))
17	16	fodjuomnilemdc 7009	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑂) → DECID ∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧))
18	17	adantrr 470	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → DECID ∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧))
19	13, 15, 18	ifcldcd 3502	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → if(∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧), ∅, 1o) ∈ ω)
20	7, 10, 11, 19	fvmptd3 5507	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → (𝑃‘𝑤) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧), ∅, 1o))
21	6, 20	eqtr3d 2172	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → ∅ = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧), ∅, 1o))
22		eqifdc 3501	. . . . . . . 8 ⊢ (DECID ∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧) → (∅ = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧), ∅, 1o) ↔ ((∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧) ∧ ∅ = ∅) ∨ (¬ ∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧) ∧ ∅ = 1o))))
23	18, 22	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → (∅ = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧), ∅, 1o) ↔ ((∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧) ∧ ∅ = ∅) ∨ (¬ ∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧) ∧ ∅ = 1o))))
24	21, 23	mpbid 146	. . . . . 6 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → ((∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧) ∧ ∅ = ∅) ∨ (¬ ∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧) ∧ ∅ = 1o)))
25	5, 24	ecased 1327	. . . . 5 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → (∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧) ∧ ∅ = ∅))
26	25	simpld 111	. . . 4 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → ∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧))
27		rexm 3457	. . . 4 ⊢ (∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧) → ∃𝑧 𝑧 ∈ 𝐴)
28	26, 27	syl 14	. . 3 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → ∃𝑧 𝑧 ∈ 𝐴)
29		eleq1w 2198	. . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
30	29	cbvexv 1890	. . 3 ⊢ (∃𝑧 𝑧 ∈ 𝐴 ↔ ∃𝑥 𝑥 ∈ 𝐴)
31	28, 30	sylib 121	. 2 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → ∃𝑥 𝑥 ∈ 𝐴)
32	1, 31	rexlimddv 2552	1 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697  DECID wdc 819   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∃wrex 2415  ∅c0 3358  ifcif 3469   ↦ cmpt 3984  ωcom 4499  –onto→wfo 5116  ‘cfv 5118  1_oc1o 6299   ⊔ cdju 6915  inlcinl 6923

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-1st 6031  df-2nd 6032  df-1o 6306  df-dju 6916  df-inl 6925  df-inr 6926

This theorem is referenced by:  fodjuomnilemres  7013  fodjumkvlemres  7026