Theorem fodju0 7451

Description: Lemma for fodjuomni 7453 and fodjumkv 7464. 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 7355	. . . . 5 ⊢ (𝑢 ∈ 𝐴 → (inl‘𝑢) ∈ (𝐴 ⊔ 𝐵))
3		foelrn 5931	. . . . 5 ⊢ ((𝐹:𝑂–onto→(𝐴 ⊔ 𝐵) ∧ (inl‘𝑢) ∈ (𝐴 ⊔ 𝐵)) → ∃𝑣 ∈ 𝑂 (inl‘𝑢) = (𝐹‘𝑣))
4	1, 2, 3	syl2an 289	. . . 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 5684	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ((𝐹‘𝑦) = (inl‘𝑧) ↔ (𝐹‘𝑣) = (inl‘𝑧)))
7	6	rexbidv 2545	. . . . . . 7 ⊢ (𝑦 = 𝑣 → (∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧) ↔ ∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧)))
8	7	ifbid 3648	. . . . . 6 ⊢ (𝑦 = 𝑣 → if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧), ∅, 1o))
9		simprl 531	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → 𝑣 ∈ 𝑂)
10		peano1 4721	. . . . . . . 8 ⊢ ∅ ∈ ω
11	10	a1i 9	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → ∅ ∈ ω)
12		1onn 6766	. . . . . . . 8 ⊢ 1o ∈ ω
13	12	a1i 9	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → 1o ∈ ω)
14	1	fodjuomnilemdc 7448	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑂) → DECID ∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧))
15	14	ad2ant2r 509	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → DECID ∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧))
16	11, 13, 15	ifcldcd 3664	. . . . . 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 5776	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → (𝑃‘𝑣) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧), ∅, 1o))
18		fveqeq2 5684	. . . . . 6 ⊢ (𝑤 = 𝑣 → ((𝑃‘𝑤) = 1o ↔ (𝑃‘𝑣) = 1o))
19		fodju0.1	. . . . . . 7 ⊢ (𝜑 → ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o)
20	19	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o)
21	18, 20, 9	rspcdva 2928	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → (𝑃‘𝑣) = 1o)
22		simplr 529	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → 𝑢 ∈ 𝐴)
23		simprr 533	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → (inl‘𝑢) = (𝐹‘𝑣))
24	23	eqcomd 2240	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → (𝐹‘𝑣) = (inl‘𝑢))
25		fveq2 5675	. . . . . . . 8 ⊢ (𝑧 = 𝑢 → (inl‘𝑧) = (inl‘𝑢))
26	25	rspceeqv 2942	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐴 ∧ (𝐹‘𝑣) = (inl‘𝑢)) → ∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧))
27	22, 24, 26	syl2anc 411	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → ∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧))
28	27	iftrued 3633	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → if(∃𝑧 ∈ 𝐴 (𝐹‘𝑣) = (inl‘𝑧), ∅, 1o) = ∅)
29	17, 21, 28	3eqtr3rd 2276	. . . 4 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ (𝑣 ∈ 𝑂 ∧ (inl‘𝑢) = (𝐹‘𝑣))) → ∅ = 1o)
30	4, 29	rexlimddv 2667	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → ∅ = 1o)
31		1n0 6678	. . . . 5 ⊢ 1o ≠ ∅
32	31	nesymi 2460	. . . 4 ⊢ ¬ ∅ = 1o
33	32	a1i 9	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → ¬ ∅ = 1o)
34	30, 33	pm2.65da 667	. 2 ⊢ (𝜑 → ¬ 𝑢 ∈ 𝐴)
35	34	eq0rdv 3557	1 ⊢ (𝜑 → 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  DECID wdc 842   = wceq 1398   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  ∅c0 3512  ifcif 3624   ↦ cmpt 4176  ωcom 4717  –onto→wfo 5355  ‘cfv 5357  1_oc1o 6653   ⊔ cdju 7341  inlcinl 7349

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-1st 6347  df-2nd 6348  df-1o 6660  df-dju 7342  df-inl 7351  df-inr 7352

This theorem is referenced by:  fodjuomnilemres  7452  fodjumkvlemres  7463