Theorem fodjum 7221

Description: Lemma for fodjuomni 7224 and fodjumkv 7235. 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 6499	. . . . . . . . 9 ⊢ 1o ≠ ∅
3	2	nesymi 2413	. . . . . . . 8 ⊢ ¬ ∅ = 1o
4	3	intnan 930	. . . . . . 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 531	. . . . . . . 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 5570	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑦) = (inl‘𝑧) ↔ (𝐹‘𝑤) = (inl‘𝑧)))
9	8	rexbidv 2498	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧) ↔ ∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧)))
10	9	ifbid 3583	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧), ∅, 1o))
11		simprl 529	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → 𝑤 ∈ 𝑂)
12		peano1 4631	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
13	12	a1i 9	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → ∅ ∈ ω)
14		1onn 6587	. . . . . . . . . . 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 7219	. . . . . . . . . . 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 3598	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → if(∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧), ∅, 1o) ∈ ω)
20	7, 10, 11, 19	fvmptd3 5658	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → (𝑃‘𝑤) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧), ∅, 1o))
21	6, 20	eqtr3d 2231	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → ∅ = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧), ∅, 1o))
22		eqifdc 3597	. . . . . . . 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 1360	. . . . 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 3551	. . . 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 2257	. . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
30	29	cbvexv 1933	. . 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 2619	1 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ∃wrex 2476  ∅c0 3451  ifcif 3562   ↦ cmpt 4095  ωcom 4627  –onto→wfo 5257  ‘cfv 5259  1_oc1o 6476   ⊔ cdju 7112  inlcinl 7120

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-1st 6207  df-2nd 6208  df-1o 6483  df-dju 7113  df-inl 7122  df-inr 7123

This theorem is referenced by:  fodjuomnilemres  7223  fodjumkvlemres  7234