Theorem bnj1542 34871

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1542.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
bnj1542.2	⊢ (𝜑 → 𝐺 Fn 𝐴)
bnj1542.3	⊢ (𝜑 → 𝐹 ≠ 𝐺)
bnj1542.4	⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹)

Assertion

Ref	Expression
bnj1542	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ≠ (𝐺‘𝑥))

Distinct variable groups:   𝑥,𝐴   𝑤,𝐹   𝑤,𝐺,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑤)   𝐴(𝑤)   𝐹(𝑥)

Proof of Theorem bnj1542

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1542.3	. . 3 ⊢ (𝜑 → 𝐹 ≠ 𝐺)
2		bnj1542.1	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		bnj1542.2	. . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐴)
4		eqfnfv 7051	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘𝑦)))
5	4	necon3abid 2977	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 ≠ 𝐺 ↔ ¬ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘𝑦)))
6		df-ne 2941	. . . . . . 7 ⊢ ((𝐹‘𝑦) ≠ (𝐺‘𝑦) ↔ ¬ (𝐹‘𝑦) = (𝐺‘𝑦))
7	6	rexbii 3094	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦) ↔ ∃𝑦 ∈ 𝐴 ¬ (𝐹‘𝑦) = (𝐺‘𝑦))
8		rexnal 3100	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 ¬ (𝐹‘𝑦) = (𝐺‘𝑦) ↔ ¬ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘𝑦))
9	7, 8	bitri 275	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦) ↔ ¬ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘𝑦))
10	5, 9	bitr4di 289	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 ≠ 𝐺 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦)))
11	2, 3, 10	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐹 ≠ 𝐺 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦)))
12	1, 11	mpbid 232	. 2 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦))
13		nfv 1914	. . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) ≠ (𝐺‘𝑥)
14		bnj1542.4	. . . . . 6 ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹)
15	14	nfcii 2894	. . . . 5 ⊢ Ⅎ𝑥𝐹
16		nfcv 2905	. . . . 5 ⊢ Ⅎ𝑥𝑦
17	15, 16	nffv 6916	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦)
18		nfcv 2905	. . . 4 ⊢ Ⅎ𝑥(𝐺‘𝑦)
19	17, 18	nfne 3043	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) ≠ (𝐺‘𝑦)
20		fveq2 6906	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
21		fveq2 6906	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐺‘𝑥) = (𝐺‘𝑦))
22	20, 21	neeq12d 3002	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ (𝐹‘𝑦) ≠ (𝐺‘𝑦)))
23	13, 19, 22	cbvrexw 3307	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦))
24	12, 23	sylibr 234	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ≠ (𝐺‘𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   Fn wfn 6556  ‘cfv 6561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569

This theorem is referenced by:  bnj1523  35085