Theorem fodjum 7476

Description: Lemma for fodjuomni 7479 and fodjumkv 7490. 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 6695	. . . . . . . . 9 ⊢ 1o ≠ ∅
3	2	nesymi 2466	. . . . . . . 8 ⊢ ¬ ∅ = 1o
4	3	intnan 941	. . . . . . 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 537	. . . . . . . 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 5699	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑦) = (inl‘𝑧) ↔ (𝐹‘𝑤) = (inl‘𝑧)))
9	8	rexbidv 2551	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧) ↔ ∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧)))
10	9	ifbid 3659	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧), ∅, 1o))
11		simprl 535	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → 𝑤 ∈ 𝑂)
12		peano1 4736	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
13	12	a1i 9	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → ∅ ∈ ω)
14		1onn 6783	. . . . . . . . . . 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 7474	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑂) → DECID ∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧))
18	17	adantrr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → DECID ∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧))
19	13, 15, 18	ifcldcd 3675	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → if(∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧), ∅, 1o) ∈ ω)
20	7, 10, 11, 19	fvmptd3 5793	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → (𝑃‘𝑤) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧), ∅, 1o))
21	6, 20	eqtr3d 2273	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑂 ∧ (𝑃‘𝑤) = ∅)) → ∅ = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑤) = (inl‘𝑧), ∅, 1o))
22		eqifdc 3674	. . . . . . . 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 1390	. . . . 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 3624	. . . 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 2299	. . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
30	29	cbvexv 1974	. . 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 2673	1 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720  DECID wdc 846   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∃wrex 2529  ∅c0 3520  ifcif 3635   ↦ cmpt 4187  ωcom 4732  –onto→wfo 5370  ‘cfv 5372  1_oc1o 6670   ⊔ cdju 7367  inlcinl 7375

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-1st 6364  df-2nd 6365  df-1o 6677  df-dju 7368  df-inl 7377  df-inr 7378

This theorem is referenced by:  fodjuomnilemres  7478  fodjumkvlemres  7489