Theorem fodju0 6931

Description: Lemma for fodjuomni 6933 and fodjumkv 6945. A condition which shows that 𝐴 is empty. (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))
fodju0.1	⊢ (𝜑 → ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o)

Assertion

Ref	Expression
fodju0	⊢ (𝜑 → 𝐴 = ∅)

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

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

Proof of Theorem fodju0

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fodjuf.fo	. . . . 5 ⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))
2		djulcl 6851	. . . . 5 ⊢ (𝑢 ∈ 𝐴 → (inl‘𝑢) ∈ (𝐴 ⊔ 𝐵))
3		foelrn 5586	. . . . 5 ⊢ ((𝐹:𝑂–onto→(𝐴 ⊔ 𝐵) ∧ (inl‘𝑢) ∈ (𝐴 ⊔ 𝐵)) → ∃𝑣 ∈ 𝑂 (inl‘𝑢) = (𝐹‘𝑣))
4	1, 2, 3	syl2an 285	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → ∃𝑣 ∈ 𝑂 (inl‘𝑢) = (𝐹‘𝑣))
5		fodjuf.p	. . . . . 6 ⊢ 𝑃 = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o))
6		fveqeq2 5362	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ((𝐹‘𝑦) = (inl‘𝑧) ↔ (𝐹‘𝑣) = (inl‘𝑧)))
7	6	rexbidv 2397	. . . . . . 7 ⊢ (𝑦 = 𝑣 → (∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧) ↔ ∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧)))
8	7	ifbid 3440	. . . . . 6 ⊢ (𝑦 = 𝑣 → if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧), ∅, 1o))
9		simprl 501	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → 𝑣 ∈ 𝑂)
10		peano1 4446	. . . . . . . 8 ⊢ ∅ ∈ ω
11	10	a1i 9	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → ∅ ∈ ω)
12		1onn 6346	. . . . . . . 8 ⊢ 1o ∈ ω
13	12	a1i 9	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → 1o ∈ ω)
14	1	fodjuomnilemdc 6928	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑂) → DECID ∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧))
15	14	ad2ant2r 496	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → DECID ∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧))
16	11, 13, 15	ifcldcd 3454	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → if(∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧), ∅, 1o) ∈ ω)
17	5, 8, 9, 16	fvmptd3 5446	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → (𝑃‘𝑣) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧), ∅, 1o))
18		fveqeq2 5362	. . . . . 6 ⊢ (𝑤 = 𝑣 → ((𝑃‘𝑤) = 1o ↔ (𝑃‘𝑣) = 1o))
19		fodju0.1	. . . . . . 7 ⊢ (𝜑 → ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o)
20	19	ad2antrr 475	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o)
21	18, 20, 9	rspcdva 2749	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → (𝑃‘𝑣) = 1o)
22		simplr 500	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → 𝑢 ∈ 𝐴)
23		simprr 502	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → (inl‘𝑢) = (𝐹‘𝑣))
24	23	eqcomd 2105	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → (𝐹‘𝑣) = (inl‘𝑢))
25		fveq2 5353	. . . . . . . 8 ⊢ (𝑧 = 𝑢 → (inl‘𝑧) = (inl‘𝑢))
26	25	rspceeqv 2761	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐴 ∧ (𝐹‘𝑣) = (inl‘𝑢)) → ∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧))
27	22, 24, 26	syl2anc 406	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → ∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧))
28	27	iftrued 3428	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → if(∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧), ∅, 1o) = ∅)
29	17, 21, 28	3eqtr3rd 2141	. . . 4 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → ∅ = 1o)
30	4, 29	rexlimddv 2513	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → ∅ = 1o)
31		1n0 6259	. . . . 5 ⊢ 1o ≠ ∅
32	31	nesymi 2313	. . . 4 ⊢ ¬ ∅ = 1o
33	32	a1i 9	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → ¬ ∅ = 1o)
34	30, 33	pm2.65da 628	. 2 ⊢ (𝜑 → ¬ 𝑢 ∈ 𝐴)
35	34	eq0rdv 3354	1 ⊢ (𝜑 → 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  DECID wdc 786   = wceq 1299   ∈ wcel 1448  ∀wral 2375  ∃wrex 2376  ∅c0 3310  ifcif 3421   ↦ cmpt 3929  ωcom 4442  –onto→wfo 5057  ‘cfv 5059  1_oc1o 6236   ⊔ cdju 6837  inlcinl 6845

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-1st 5969  df-2nd 5970  df-1o 6243  df-dju 6838  df-inl 6847  df-inr 6848

This theorem is referenced by:  fodjuomnilemres  6932  fodjumkvlemres  6944