Theorem fodjum 7344

Description: Lemma for fodjuomni 7347 and fodjumkv 7358. 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 6599	. . . . . . . . 9 ⊢ 1o ≠ ∅
3	2	nesymi 2448	. . . . . . . 8 ⊢ ¬ ∅ = 1o
4	3	intnan 936	. . . . . . 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 533	. . . . . . . 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 5648	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑦) = (inl‘𝑧) ↔ (𝐹‘𝑤) = (inl‘𝑧)))
9	8	rexbidv 2533	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧) ↔ ∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧)))
10	9	ifbid 3627	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧), ∅, 1o))
11		simprl 531	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → 𝑤 ∈ 𝑂)
12		peano1 4692	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
13	12	a1i 9	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → ∅ ∈ ω)
14		1onn 6687	. . . . . . . . . . 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 7342	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑂) → DECID ∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧))
18	17	adantrr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → DECID ∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧))
19	13, 15, 18	ifcldcd 3643	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → if(∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧), ∅, 1o) ∈ ω)
20	7, 10, 11, 19	fvmptd3 5740	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → (𝑃‘𝑤) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧), ∅, 1o))
21	6, 20	eqtr3d 2266	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → ∅ = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧), ∅, 1o))
22		eqifdc 3642	. . . . . . . 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 147	. . . . . 6 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → ((∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧) ∧ ∅ = ∅) ∨ (¬ ∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧) ∧ ∅ = 1o)))
25	5, 24	ecased 1385	. . . . 5 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → (∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧) ∧ ∅ = ∅))
26	25	simpld 112	. . . 4 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → ∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧))
27		rexm 3594	. . . 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 2292	. . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
30	29	cbvexv 1967	. . 3 ⊢ (∃𝑧 𝑧 ∈ 𝐴 ↔ ∃𝑥 𝑥 ∈ 𝐴)
31	28, 30	sylib 122	. 2 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → ∃𝑥 𝑥 ∈ 𝐴)
32	1, 31	rexlimddv 2655	1 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 715  DECID wdc 841   = wceq 1397  ∃wex 1540   ∈ wcel 2202  ∃wrex 2511  ∅c0 3494  ifcif 3605   ↦ cmpt 4150  ωcom 4688  –onto→wfo 5324  ‘cfv 5326  1_oc1o 6574   ⊔ cdju 7235  inlcinl 7243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1st 6302  df-2nd 6303  df-1o 6581  df-dju 7236  df-inl 7245  df-inr 7246

This theorem is referenced by:  fodjuomnilemres  7346  fodjumkvlemres  7357